2. Working with Linear Equations
3. Working with Inequalities and Absolute Values
4. Working with Linear Systems
To solve tough SAT math problems, you must be fluent in defining, manipulating, and analyzing algebraic expressions.
Corrine drives to her office at an average speed of 50 miles per hour. When she returns home by the same route, the traffic is lighter and she averages 60 miles per hour. If her trip home is 10 minutes shorter than her trip to her office, what is the distance, in miles, from Corrine’s home to her office?
(Medium-hard) Why does everyone hate “word problems” like this one? For most of us, the problem is that the equations aren’t “set up” for us—we have to set them up ourselves, which can be a pain in the neck. But we can make these problems much easier by breaking them down into clear steps.
Step 1. Identify: In this problem, there are six relevant quantities:
• the speed from home to work
• the distance from home to work
• the time it takes to get from home to work
• the speed from work to home
• the distance from work to home
• the time it takes to get from work to home
This may seem like a lot, but as we will see, keeping track of them is quite manageable.
Step 2. Express: The problem gives us enough information to express all six quantities in terms of only two “unknowns.” If d is the distance, in miles, from her home to her office, and t is the time, in hours, it takes her to get home from the office, then we can express our six quantities, respectively, as
Step 3. Translate: To translate the facts of this problem into equations, we must know the formula distance = average speed × time. Applying this to each trip gives us
Step 4. Analyze and Solve: We have now reduced the problem to a “two by two system,” that is, two equations with two unknowns. Since the number of equations equals the number of unknowns, we should be able to solve for those unknowns. (In Lessons 12 and 13, we will review these concepts and techniques.) Since the unknown d is isolated in both equations, substitution is simple:
Since t represents the time it took Corrine to return home, in hours, this means it took her 5/6 hours (or 50 minutes) to get from her office to her home, and 5/6 hour + 1/6 hour = 1 hour to get to her office from home. But remember, the question asks for the distance from her home to her office, which we can find by substituting into either of our equations:
50(5/6 + 1/6) or 60(5/6) = 50 miles
When expressing or simplifying a quantity, you frequently have many options. For instance, the expression 4x2 – 12x can also be expressed as 4x(x – 3). Similarly, 3.2 can be expressed as 16/5 or 3 ⅕ or 32/10. Which way is better? It depends on what you want to do with the expression. Different forms of an expression can reveal different characteristics of that quantity or the equation in which it appears. To gain fluency in expressing quantities, you must understand the Laws of Arithmetic.
To simplify complex expressions, you must know the Order of Operations:
PG-ER-MD-AS
Step 1: PG (parentheses and other grouping symbols, from innermost to outermost and left to right)
Since this expression contains no parentheses, we don’t have to worry about “grouped” operations, right? Wrong! Remember that fraction bars and radicals are “grouping symbols” just like parentheses are.
In other words, we can think of this expression as
If a set of parentheses contains only one operation, then we simply do that operation:
If it contains more than one operation, then we must move on to the next step.
Step 2: ER (exponents and roots, from innermost to outermost and left to right)
Do any of the parentheses contain exponents or roots? Yes, so we must perform that operation next:
Step 3: MD (multiplication and division, from left to right)
Next, we do any multiplication inside the parentheses:
Step 4: AS (addition and subtraction, from left to right)
Now we do any addition and subtraction left in the parentheses:
Once all the “grouped” operations are completed, we run through the order of operations once again to finish up. Exponents or roots? No. Multiplication or division? Yes:
1.875 + 2
Addition or subtraction? Yes: 1.875 + 2 = 3.875
What is the sum of the first 100 positive integers?
(Hard) Here, following the order of operations would be, shall we say, less than convenient: it would require 99 computations. Even with a calculator, it would be a pain. But here is a much simpler method:
Original expression:
1 + 2 + 3 + 4 + … + 97 + 98 + 99 + 100
Rearrange and regroup:
(1 + 100) + (2 + 99) + (3 + 98) + … + (50 + 51)
Simplify:
(101) + (101) + (101) + … + (101)
Since we have 50 pairs, this equals:
50(101)
Simplify:
5,050
This gives us exactly the same result as the order of operations would give, but with just a few simple calculations. How did we do it? By using three more laws of arithmetic: the commutative law of addition, the associative law of addition, and the distributive law of multiplication over addition.
Which of the following is equivalent to 3(34 × 53)? [No calculator]
A) 3(34) × 3(53)
B) 94 + 153
C) 94 × 153
D) 35 × 53
Before making your choice, check the laws of arithmetic; don’t make up your own laws. Which laws of arithmetic can we use? Since the expression is a product, we can use the commutative law of multiplication and jumble up the terms, or the associative law of multiplication and regroup the terms any way we want (or not at all). Using the associative law gives us
If x ≠ 0, which of the following equals 
?
A) 
B) 
C) 2x2 + 2
D) 4x + 2
(Medium)
Commutative Law of Addition:
Associative Law of Addition:
Distributive Law:
Division by a number is multiplication by its reciprocal:
Distributive Law:
So the correct answer is (D). Look at each step carefully and notice how each one uses a particular Law of Arithmetic. In particular, notice that the “combining of like terms” in steps 1–3 is really an example of commuting, associating, and (un)distributing. Even more interesting, notice that steps 4–5 show that division distributes just like multiplication does.
You can also distribute division over addition just as you can distribute multiplication.
How many distinct values of x are solutions to the equation x2 + 4 = −4x?
A) none
B) one
C) two
D) three
(Medium) You might recognize that this equation is a quadratic equation (which we will discuss in much more detail in Chapter 9) and remember that such equations usually have two distinct solutions, but not always, so we must look at this equation more carefully.
Step 2 might seem a bit mysterious. Why did we write 4x as 2x + 2x? Here we are using the Product-Sum Method for factoring quadratics, which is explained in a bit more detail in Chapter 9, Lesson 4. For now, though, just notice that each step follows a particular Law of Arithmetic.
If the product of two numbers is 0, then one of those numbers must be 0. (This is the Zero Product Property.) Therefore x + 2 = 0 and so x = −2. Since the other factor is the same, we only get one solution to this equation, and the answer is (B).
To check the equation in step 5, we can FOIL the product of binomials on the left side to make sure we get the same expression we had back in step 1: (x + 2)(x + 2) = x2 + 4x + 4, which is precisely the expression we started with in step 1, confirming that our work is correct.
This means that the factoring process in steps 2–5 can be thought of as un-FOILing. We will look at this method of factoring more carefully in Chapter 9.
Make sure you know how to FOIL and un-FOIL.
FOILing is simply the shortcut for multiplying two binomials, which requires applying the distributive law twice. For example:
(1 − (1 − (1 − 2))) − (1 − (1 − (1 − 3))) =
When 14 is subtracted from 6 times a number, 40 is left. What is half the number?
Four consecutive even numbers have a sum of 76. What is the greatest of these numbers?
If 
, then 10x + 12 =
What number decreased by 7 equals the opposite of five times the number?
If 5d + 12 = 24, then 5d − 12 =
If 
, then y + 5 =
The product of x and y is 36. If both x and y are integers, then what is the least possible value of x − y?
A) −37
B) −36
C) −35
D) −9
If a factory can manufacture b computer screens in n days at a cost of c dollars per screen, then which of the following represents the total cost, in dollars, of the computer screens that can be manufactured, at that rate, in m days?
A) 
B) 
C) 
D) 
Which of the following is equivalent to 5x(2x × 3) − 5x2 for all real values of x?
A) 5x2 + 15x
B) 25x2
C) 5x2 − 15x
D) 10x2 × 15x − 5x2
The symbol Ο represents one of the fundamental operators: +, −, ×, or ÷. If (x Ο y) × (y Ο x) = 1 for all positive values of x and y, then Ο can represent
A) +
B) ×
C) −
D) ÷
The difference of two numbers is 4 and their sum is 14. What is their product?
If x + y − 1 = 1 − (1 − x), what is the value of y?
If 3x2 + 2x = 40, then 15x2 + 10x =
Ellen is currently twice as old as Maria, but in 6 years, Maria will be 2/3 as old as Ellen. How old is Ellen now?
If 2x − 2y = 5 and x + y = 6, what is the value of x2 − y2?
On a typical day, a restaurant sells n grilled cheese sandwiches for p dollars each. Today, however, the manager reduced the price of grilled cheese sandwiches by 30% and as a result sold 50% more of them than usual. Which of the following represents the revenue for today’s grilled cheese sandwich sales, in dollars?
A) 0.5np − 0.3
B) 1.05np
C) 1.20np
D) 1.50np
For all real numbers x and y, 4x(x) − 3xy(2x) =
A) 12x2y(x − 2y)
B) 2x2(2 − 3y)
C) 2x2(2 + 3y)
D) 4xy(x − 3y)
If a = 60(99)99 + 30(99)99, b = 99100, and c = 90(90)99, then which of the following expresses the correct ordering of a, b, and c?
A) c < b < a
B) b < c < a
C) a < b < c
D) c < a < b
Which of the following statements must be true for all values of x, y, and z?
I. (x + y) + z = (z + y) + x
II. (x − y) − z = (z − y) − x
III. (x ÷ y) ÷ z = (z ÷ y) ÷ x
A) I only
B) I and II only
C) I and III only
D) II and III only
Carlos began with twice as much money as David had. After Carlos gave $12 to David, Carlos still had $10 more than David. How much money did they have combined at the start?
A) $34
B) $68
C) $102
D) $108
1. 1
(1 − (1 − (1 − 2))) − (1 − (1 − (1 − 3)))
Parentheses:
(1 − (1 − (−1))) − (1− (1 − (−2)))
Next parentheses:
(1 − (2)) − (1 − (3))
Next parentheses:
(−1) − (−2)
Subtract:
−1 + 2 = 1
2. 9/2 or 4.5
6x − 14 = 40
Add 14:
6x = 54
Divide by 6:
x = 9
Multiply by 
:
3. 22 Let n be the least of these numbers. The sum of four consecutive even numbers is therefore n + (n + 2) + (n + 4) + (n + 6) = 76.
Simplify:
4n + 12 = 76
Subtract 12:
4n = 64
Divide by 4:
n = 16
Therefore the largest of these numbers is 16 + 6 = 22.
4. 28
Multiply by 4:
10x + 12 = 28
5. 7/6 or 1.16 or 1.17
x − 7 = −5x
Subtract x:
−7 = −6x
Divide by –6:
6. 0
5d + 12 = 24
Subtract 24:
5d − 12 = 0
7. 5
Subtract y2:
Multiply by −5/3:
y2 = 0
Take square root:
y = 0
Add 5:
y + 5 = 5
8. C If xy = 36 and x and y are integers, then x and y are both factors of 36. In order to minimize the value of x − y, we must find the greatest separation between x and y. The greatest separation between a factor pair is 1 − 36 = −35.
9. A We should regard this as a “conversion” problem from m days into a corresponding number of dollars.
10. B Original expression:
5x(2x × 3) − 5x2
Parentheses:
5x(6x) − 5x2
Multiply:
30x2 − 5x2
Subtract:
25x2
Remember: The Law of Distribution does not apply in the first step, because the grouped expression doesn’t include addition or subtraction.
11. D The simplest approach is perhaps to choose simple values for x and y, like 2 and 3, and see which operator yields a true equation. Since (2 ÷ 3) × (3 ÷ 2) = 1, the answer is (D).
12. 45
a − b = 4
a + b = 14
Add equations:
2a = 18
Divide by 2:
a = 9
Substitute a = 9:
9 + b = 14
Subtract 9:
b = 5
Evaluate ab:
ab = 9 × 5 = 45
13. 1
x + y − 1 = 1 − (1 − x)
Distribute:
x + y − 1 = 1 − 1 + x
Subtract x:
y − 1 = 1 − 1
Simplify:
y − 1 = 0
Add 1:
y = 1
14. 200
3x2 + 2x = 40
Multiply by 5:
15x2 + 10x = 200
15. 12 Let e = Ellen’s current age and m = Maria’s current age.
Ellen is twice as old as Maria:
e = 2m
In 6 years, Maria will be 2/3 as old as Ellen:
Substitute e = 2m:
Multiply by 3:
3m + 18 = 2(2m + 6)
Distribute:
3m + 18 = 4m + 12
Subtract 3m and 12:
6 = m
Therefore e = 2m = 2(6) = 12.
16. 15 First equation:
2x − 2y = 5
Divide by 2:
x − y = 2.5
Second equation:
x + y = 6
Multiply:
(x − y)(x + y) = x2 − y2 = (2.5)(6) = 15
Alternately, we could solve the system using either substitution or linear combination and get x = 4.25 and y = 1.75, and evaluate x2 − y2 = (4.25)2 − (1.75)2 = 18.0625 − 3.0625 = 15.
17. B The revenue is equal to the number of items sold times the price per item. If the restaurant typically sells n sandwiches per day, but today sold 50% more, it sold 1.5n sandwiches. If the price p was reduced 30%, today’s price is 0.70p. Therefore, the total revenue is (1.5n)(0.70p) = 1.05np.
18. B
4x(x) − 3xy(2x)
Multiply:
4x2 − 6x2y
Largest common factor:
2x2(2 − 3y)
19. D Although a calculator is permitted for this question, most calculators will give an “overflow error” when trying to calculate numbers like 99100, because they’re just too large. However, comparing these numbers is straightforward if we can express them in a common format.
20. A Only statement I is true, by the Commutative and Associative Laws of Addition. Choosing simple values like x = 1, y = 2, and z = 3 will demonstrate that statements II and III do not yield true equations.
21. C Let x = the number of dollars David had to start. If Carlos started with twice as much money as David, then Carlos started with 2x dollars. After Carlos gave David $12, Carlos had 2x − 12 dollars and David had x + 12 dollars. If Carlos still had $10 more than David, then
2x − 12 = 10 + x + 12
Simplify:
2x − 12 = x + 22
Add 12:
2x = x + 34
Subtract x:
x = 34
Therefore, David started with $34 and Carlos started with 2($34) = $68, so they had $34 + $68 = $102 combined to start.
If x and y are positive numbers and 3x − 2y = 7, what is the value of 
?
A) 
B) 
C) 
D)
(Easy) Working with algebraic equations doesn’t always mean “solving for x.” Notice that this particular question doesn’t ask for the values of x or y, but rather for the value of a more complicated expression. This may seem harder, but it’s actually pretty simple if you understand the Law of Substitution
How does this help us here? Notice that if we simply add 2y to both sides of the equation, we get
3x − 2y = 7
Add 2y:
3x = 2y + 7
Therefore, by the Law of Substitution, we can substitute 3x for 2y + 7 or vice-versa. Since 2y + 7 appears in the expression we are asked to evaluate, it makes sense to replace it with 3x:
Substitute 3x for 2y + 7:
Simplify:
Increasing a positive number x by 25% and then decreasing the result by 50% is equivalent to dividing x by what number?
A) 1.333
B) 1.5
C) 1.6
D) 1.625
(Medium) Increasing a quantity by 25% is equivalent to multiplying it by 1.25, because the final amount is 125% of the original amount (Chapter 8, Lesson 8). Decreasing a quantity by 50% is equivalent to multiplying it by .5, because the final amount is 50% of the original amount. Therefore, performing both changes is equivalent to multiplying by 1.25 × 0.50, or 0.625, which is equal to 5/8. But the question asks us for the equivalent division. Here, we need to remember a simple rule: multiplying by a number is equivalent to dividing by its reciprocal. Therefore, multiplying by 5/8 is the same as dividing by 8/5, which is 1.6. Therefore, the correct answer is (C).
Every operation can be expressed in terms of its inverse operation.
And here are two more handy equivalences:
If 
what is the value of m + n?
(Easy) When a problem includes a complicated expression, we should try to simplify it, but always keep an eye on what the question is asking. In this case, simplifying to find the value of m + n requires knowing some factoring identities.
The first of these identities helps us factor our numerator:
Factor numerator and denominator:
Cancel common factors:
Multiply by 2:
m + n = 9
Niko is 27 inches shorter than his father, who is 5 feet 10 inches tall. How tall is Niko? (1 foot = 12 inches)
A) 3 feet 4 inches
B) 3 feet 6 inches
C) 3 feet 7 inches
D) 3 feet 10 inches
(Easy) Solving this problem requires unit conversions. To convert inches to feet, we multiply by the conversion factor (1 foot/12 inches). To convert feet to inches, we multiply by its reciprocal (12 inches/1 foot). If Niko’s father is 5 feet 10 inches tall, he is 5 feet × (12 inches/1 foot) + 10 inches = 70 inches tall. If Niko is 27 inches shorter, he is 70 − 27 = 43 inches tall, which is equivalent to 43 inches × (1 foot/12 inches) = 3 7/12 feet, or 3 feet 7 inches, so the correct answer is (C).
If a factory can manufacture b computer screens in n days at a cost of c dollars per screen, then which of the following represents the total cost, in dollars, of the computer screens that can be manufactured, at that rate, in m days?
A) 
B) 
C) 
D) 
(Medium) This problem, from the previous exercise set, can be solved in several different ways. One method is to simply choose values for the unknowns and turn the problem into an arithmetic problem instead of an algebra problem. But here we will look at it as a conversion problem.
We can think of this problem as being a “conversion” from a quantity of days to an equivalent quantity of dollars. We are given that this factory is working for m days, so we write this quantity down, including the units, and we multiply by the conversion factors until we get dollars:
So the correct answer is (A).
If bag A weighs 4 pounds 5 ounces and bag B weighs 6 pounds 2 ounces, how much heavier, in ounces, is bag B than bag A? (1 pound = 16 ounces)
If 
, what is the value of 
?
If x − 2y = 10 and x ≠ 0, what is the value of 
?
If a − b = 4 and a2 − b2 = 3, what is the value of a + b?
If 6 gricks are equivalent to 5 merts, then 2 merts are equivalent to how many gricks?
If the function {x} is defined by the equation {x} = (1 − x)2, what is the value of {{4}}?
If 
and 
, what is the value of 
?
(x − 9)(x − a) = x2 − 4ax + b
In the equation above, a and b are constants. If the equation is true for all values of x, what is the value of b?
A) −27
B) −12
C) 12
D) 27
If 
, what is the value of x?
A) 
B) −7
C) 
D) 
(p + 2)2 = (p − 5)2
The equation above is true for which of the following values of p?
A) −2 and 5
B) 2 and −5
C) 1.5 only
D) 5 only
If 
for all positive values of m and n, then which of the following is equal to x?
A) 
B) 
C) 
D) 
Let m be a positive real number. Increasing m by 60% then decreasing the result by 50% is equivalent to dividing m by what number?
What is the sum of the first 50 positive even integers?
Three years ago, Nora was half as old as Mary is now. If Mary is four years older than Nora, how old is Mary now?
If 2/3 of the seats at a football stadium were filled at the beginning of the game, and at halftime 1,000 spectators left, leaving 3/7 of the seats filled, what is the total number of seats in the stadium?
If three candy bars and two gumdrops cost $2.20, and four candy bars and two gumdrops cost $2.80, what is the cost, in dollars, of one gumdrop?
If 
, what is the value of x −1?
Subtracting 3 from a number and then multiplying this result by 4 is equivalent to multiplying the original number by 4 and then subtracting what number?
In a poker game, a blue chip is worth 2 dollars more than a red chip, and a red chip is worth 2 dollars more than a green chip. If 5 green chips are worth m dollars, then which of the following represents the value, in dollars, of 10 blue chips and 5 red chips?
A) 50 + 3m
B) 18 + 60m
C) 40 + 3m
D) 28 + 20m
A train travels at an average speed of 50 miles per hour for the first 100 miles of a 200-mile trip, and at an average of 75 miles per hour for final 100 miles. What is the train’s average speed for the entire trip?
A) 58.5 mph
B) 60.0 mph
C) 62.5 mph
D) 63.5 mph
Which of the following is equivalent to 3m(m2 × 2m) for all real values of m?
A) 3m2 + 6m
B) 3m2 × 6m
C) 3m3 × 6m2
D) 6m4
If the cost of living in a certain city increased by 20% in the 10 years from 1980 to 1990, and increased by 50% in the 20 years from 1980 to 2000, what was the percent increase in the cost of living from 1990 to 2000?
A) 15%
B) 20%
C) 25%
D) 30%
1. 29 4 pounds 5 ounces = 4(16) + 5 = 69 ounces, and 6 pounds 2 ounces = 6(16) + 2 = 98 ounces. Therefore, bag B weighs 98 − 69 = 29 ounces more.
2. 2/15 or .133
Distribute division:
Simplify:
Subtract 1:
Divide by 3:
3. 4 Expression to be evaluated:
Given equation:
x − 2y = 10
Add 2y:
x = 2y + 10
Substitute x = 2y + 10:
Simplify:
Factor and simplify:
4. ¾ or .75
a2 − b2 = 3
Factor:
(a − b)(a + b) = 3
Substitute a − b = 4:
4(a + b) = 3
Divide by 4:
5. 12/5 or 2.4
6. 64
{4} = (1 − 4)2 = (−3)2 = 9
{{4}} = (1 − {4})2 = (1 − 9)2 = (−8)2 = 64
7. 2 Given equation:
Distribute division:
Subtract 1:
Reciprocate:
Given equation:
Distribute division:
Subtract 1:
Multiply:
8. D Given:
(x − 9)(x − a) = x2 − 4ax + b
FOIL:
x2 − ax − 9x + 9a = x2 − 4ax + b
Simplify:
x2 − (a + 9)x + 9a = x2 − 4ax + b
If this equation is true for all x, then the coefficients of corresponding terms must be equal, so
a + 9 = 4a
Subtract a:
9 = 3a
Divide by 3:
3 = a
Therefore b = 9a = 9(3) = 27.
9. A Given equation:
Multiply by 5x:
25 + 7x = 5x
Subtract 7x:
25 = −2x
Divide by −2:
10. C Given equation:
(p + 2)2 = (p − 5)2
FOIL:
p2 + 4p + 4 = p2 − 10p + 25
Subtract p2:
4p + 4 = −10p + 25
Add 10p:
14p + 4 = 25
Subtract 4:
14p = 21
Divide by 14:
p = 1.5
11. D Given equation:
Multiply by m − nx:
3x = 2(m − nx)
Distribute:
3x = 2m − 2nx
Add 2nx:
3x + 2nx = 2m
Factor out x:
x(3 + 2n) = 2m
Divide by 3 + 2n:
12. 1.25 Increasing a number by 60% is equivalent to multiplying it by 1.60, and decreasing a number by 50% is equivalent to multiplying it by 0.50. Therefore, performing both changes in succession is equivalent to multiplying by 1.60 × 0.50 = 0.80. Multiplying by 0.80 is equivalent to dividing by its reciprocal: 1/(0.80) = 1.25.
13. 2,550 The sum of the first 50 positive even integers is 2 + 4 + 6 + 8 + … + 100. As with the example is Lesson 2, these numbers can be regrouped into 25 pairs of numbers each of which has a sum of 2 + 100 = 102. Therefore, their sum is 25(102) = 2,550.
14. 14 Let n = Nora’s age now, and m = Mary’s age now. If 3 years ago, Nora was half as old
as Mary is now:
If Mary is 4 years older than Nora:
m = 4 + n
Subtract 4:
m − 4 = n
Substitute n = m − 4:
Simplify:
Multiply by 2:
2m − 14 = m
Subtract m and add 14:
m = 14
15. 4,200 Let x = the total number of seats in the stadium.
Subtract 
x:
Add 1,000:
Combine like terms:
Multiply by 
:
16. 0.20 Let g = the cost, in dollars, of one gumdrop, and c = the cost, in dollars, of one candy bar.
4c + 2g = 2.80
3c + 2g = 2.20
Subtract:
c = 0.60
Substitute c = 0.60:
4(0.60) + 2g = 2.80
Simplify:
2.40 + 2g = 2.80
Subtract 2.40:
2g = 0.40
Divide by 2:
g = 0.20
17. 6
Factor:
Multiply by −1:
Simplify:
Multiply by 2:
x − 1 = 6
18. 12 We can just choose a number to work with, like 10. If we subtract 3 from this number then multiply the result by 4, we get 4(10 − 3) = 28. If we multiply it by 4 and then subtract a mystery number, we get 4(10) − x = 40 − x.
28 = 40 − x
Subtract 40:
−12 = −x
Multiply by −1:
12 = x
19. A If 5 green chips are worth m dollars, then each green chip is worth m/5 dollars. If a red chip is worth 2 dollars more than a green chip, then each red chip is worth m/5 + 2 dollars. If each blue chip is worth 2 dollars more than a red chip, then each blue chip is worth m/5 + 4 dollars. Therefore, 10 blue chips and 5 red chips are worth 10(m/5 + 4) + 5(m/5 + 2) = 2m + 40 + m + 10 = 3m + 50 dollars.
20. B The average speed is equal to the total distance divided by the total time. The total distance is 200 miles. The time for the first hundred miles is (100 miles/50 mph) = 2 hours, and the time for the second hundred miles is (100 miles/75 mph) = 4/3 hours. Therefore the total time of the trip is 2 + 4/3 = 10/3 hours, and the average speed is
21. D
3m(m2 × 2m)
Parentheses:
3m(2m3)
Multiply:
6m4
22. C Assume the cost of living in 1980 was $100. If this increased by 20% from 1980 to 1990, then the cost of living in 1990 was 1.20($100) = $120. If the increase from 1980 to 2000 was 50%, then the cost of living in 2000 was 1.50($100) = $150. The percent increase from 1990 to 2000 is therefore
The Horizon Resort charges $150 per night for a single room, and a one-time valet parking fee of $35. There is a 6.5% state tax on the room charges, but no tax on the valet parking fee. Which of the following equations represents the total charges in dollars, c, for a single room, valet parking, and taxes, for a stay of n nights at The Horizon Resort?
A) c = (150 + 0.065n) + 35
B) c = 1.065(150n) + 35
C) c = 1.065(150n + 35)
D) c = 1.065(150 + 35)n
(Medium) This question asks us explicitly to set up an equation to express a mathematical relationship in a word problem. Usually, this is just the first step in analyzing the situation more deeply, for instance, finding particular values of the variables that satisfy certain conditions, or interpreting the meanings of terms or coefficients in the equation, but this problem only asks us to set up the equation.
When translating verbal information into an equation, it’s helpful to take small steps. First, since the room charge is $150 per night, the charge for n nights is $150n. If a 6.5% tax is added to this, the room charge becomes 150n + 0.065(150n) = 1.065(150n). The $35 valet parking charge is added separately, and not taxed, so the total charges are 1.065(150n) + 35, and the correct answer is (B). Notice that this equation shows a linear relationship between c and n.
Which of the following represents the equation of the line with an x-intercept of 6 that passes through the point (4, 4)?
A) 
B) y = 2x − 4
C) y = −2x + 12
D) y = −2x + 6
(Easy) This question asks you to construct the equation of a line given some facts about its graph. Start by drawing a graph (on the xy-plane) of the given information in the space next to the question. It also helps to know something about the different forms of linear equations and what they reveal about the graph of the line.
Graph of a line in the xy-plane
In this problem, we are given two points on the line: (4, 4) and the x-intercept (6, 0). We can calculate the slope using the slope formula above: slope = (4 − 0)/(4 − 6) = (4)/(−2) = −2. If we use this slope and the point (6, 0), we can set up the equation in point-slope form:
Point-slope form of equation:
y − 0 = −2(x − 6)
Simplify and distribute:
y = −2x + 12
This is the equation in (C). Notice that this equation is in slope-intercept form, and reveals that this line also has a y-intercept of 12. Check this fact against your diagram, and also check that both given points, (4, 4) and (6, 0), satisfy this equation.
If 
, what is the value of x?
A) 
B) 
C) 
D) 
(Medium) At first glance, this doesn’t look like a linear equation. But one simple move reveals that it is:
Multiply both sides by 3x:
Distribute and simplify:
3 + 2 = 12x
Divide by 12:
5/12 = x
As this shows, sometimes solving equations requires a clever use of the Laws of Equality.
If x2 = y2, then which of the following must be true?
I. x = y
II. 
III. x = |y|
A) none
B) I only
C) I and II only
D) I, II and III
(Medium-hard) This question tests your skills of deductive logic. Notice it is not asking which statements can be true, but rather which must be true. It seems that if we “unsquare” both sides of the original equation, we get the equation in I. If we divide the original equation by x on both sides, we get the equation in II. Does this mean that statements I and II are necessarily true? No, because we violated rule 3 in both cases. If x2 = y2, it does not follow that x = y. Notice that x could be 2 and y could be −2. These values certainly satisfy the original equation, but they do not satisfy the equations in I or III. They do, however, satisfy the equation in II, because 2 = (−2)2/2. However, statement II is still not necessarily true. What if x and y were both 0? This would satisfy the original equation, but 0 ≠ (0)2/(0) because 0/0 is undefined. Therefore, the correct answer is (A).
If m is a constant greater than 1, which of the following could be the graph in the xy-plane of x + my + m = 0?
A)
B)
C)
D)
(Medium-hard) First, we should try to get the equation into a more useful form. Let’s try the slope-intercept (y = mx + b) form:
x + my + m = 0
Subtract x and m:
my = −x − m
Divide by m:
This shows that the line has a slope of −1/m and a y-intercept of −1. Since the problem tells us that m is greater than 1, we know that the slope (−1/m) must be between −1 and 0. The only graph that satisfies these conditions is (B).
The points A(10, 4) and B(−2, k) are 13 units apart. Which of the following equations could describe the line that contains points A and B?
A) 13x + 12y = 178
B) 5x + 12y = 98
C) 5x − 12y = 98
D) 5x − 13y = −2
(Hard) Drawing a diagram will help us analyze this problem. Although we don’t know precisely where point B is, we know it is somewhere on the line x = −2. This gives us the following picture:
Next, notice that all of the equations given in the choices are in “standard” form, and in standard form the slope of the line is −a/b. Therefore, the slopes of these lines are, respectively, (A) −13/12, (B) −5/12, (C) 5/12, and (D) 5/13. Therefore, finding the slope of the line should help us choose the correct equation. Looking at the diagram more closely, notice that it includes two right triangles, and we can find the missing side of each one using the Pythagorean Theorem, or just by noticing that they are both 5-12-13 right triangles (52 + 122 = 132). Putting this information into the diagram shows us that B can therefore be at (−2, 9) or (−2, −1).
Therefore, the slope (rise/run) of the line containing B1 is −5/12, and the slope of the line containing B2 is 5/12. This means that our answer is either (B) or (C). How do we choose between them? Just remember that the line must contain the point (10, 4). If you plug x = 10 and y = 4 into these equations, only (B) works: 5(10) + 4(12) = 98.
If x − 2(1 − x) = 5, what is the value of x?
If f(x) = −2x + 8, and f(k) = −10, what is the value of k?
What is the slope of the line that contains the points (−2, 3) and (4, 5)?
What is the slope of the line described by the equation 
?
Line l is perpendicular to the line described by the equation 5x + 11y = 16. What is the slope of line l?
If 
, what is the value of x?
What is the y-intercept of the line containing the points (3, 7) and (6, 3)?
In the xy-plane, the graph of y = h(x) is a line with slope −2. If h(3) = 1 and h(b) = −9, what is the value of b?
If a train maintains a constant speed of 60 miles per hour, it can travel 4 miles per gallon of diesel fuel. If this train begins a trip with a full 200 gallon tank of diesel fuel, and maintains a speed of 60 miles per hour, which of the following equations represents the number of gallons, g, left in the tank t hours into the trip?
A) 
B) 
C) g = 200 −15t
D) 
The points A(2, 3) and B(m, 11), are 10 units apart. Which of the following equations could describe the line that contains points A and B?
A) 8x + 6y = 11
B) 8x − 6y = −2
C) 6x + 8y = 36
D) 6x − 8y = −12
The figure above shows a right triangle with vertices at the origin, (5, 6) and (k, 0). What is the value of k?
A) 
B) 
C) 
D) 
If the points (2, 4), (5, k), and (8, 20) are on the same line, what is the value of k?
Line l has a slope of 3 and a y-intercept of −4. What is its x-intercept?
If f(−1) = 1 and f(3) = 2 and f is a linear function, what is the slope of the graph y = f(x)?
If f(−1) = 1 and f(3) = 2 and f is a linear function, what is f(5)?
In the xy-plane, the graph of line n has an x-intercept of 2b and an y-intercept of −8b, where b ≠ 0. What is the slope of line n?
If 
, what is the value of x?
If the line 3x − 2y = 12 is graphed in the xy-plane, what is its x-intercept?
If the graphs of the equations 5x − 2y = 5 and 6x + ky = 9 are perpendicular, what is the value of k?
The net profit for the sales of a product is equal to the total revenue from the sales of that product minus the total cost for the sales of that product. If a particular model of calculator sells for $98, and the cost for making and selling n of these calculators is $(35n + 120,000), which of the following equations expresses the net profit in dollars, P, for making and selling n of these calculators?
A) P = 63n − 120,000
B) P = 63n + 120,000
C) P = 63(n − 120,000)
D) P = 63(n + 120,000)
Which of the following represents the equation of the line with an x-intercept of 5 and a y-intercept of 6?
A) 
B) 
C) 
D) 
The table above shows several ordered pairs corresponding to the linear function f. What is the value of a + b?
A) 12
B) 16
C) 20
D) It cannot be determined from the information given.
1. 7/3 or 2.33
x − 2(1 − x) = 5
Distribute:
x − 2 + 2x = 5
Simplify:
3x − 2 = 5
Add 2:
3x = 7
Divide by 3:
x = 7/3
2. 9
f(k) = −2k + 8 = −10
Subtract 8:
−2k = −18
Divide by −2:
k = 9
3. 1/3 or .333
4. 10/3 or 3.33
Multiply by 2xy:
Simplify:
2y + y = 10x
Simplify:
3y = 10x
Divide by 3:
5. 11/5 or 2.2 The slope of the given line is −5/11, so the slope of the line perpendicular to it is 11/5.
6. 9/5 or 1.8
Multiply by 10:
(x + 1) + 4x = 10
Simplify:
5x + 1 = 10
Subtract 1:
5x = 9
Divide by 5:
x = 9/5
7. 11 There are a variety of ways of solving this problem, but perhaps the simplest is to draw a quick sketch:
Notice that to get from (6, 3) to (3, 7) we must go left 3 units and up 4 units (in other words, the slope is −4/3). If we simply repeat this from (3, 7), we arrive at the y-intercept, which is (0, 11).
8. 8 This line has a slope of −2 and contains the points (3, 1) and (b, −9). Therefore 
Simplify:
Multiply by b − 3:
−2b + 6 = −10
Subtract 6:
−2b = −16
Divide by −2:
b = 8
9. C Since the tanks starts with 200 gallons, the amount it has left is 200 − the number of gallons used. The number of gallons used is
10. B Once again, a quick sketch can be very helpful. Notice that traveling from point
A(2, 3) to point B(m, 11) requires going up 8 units and right (or left) some unknown distance b. We can find b with the
Pythagorean Theorem:
82 + b2 = 102
Simplify:
64 + b2 = 100
Subtract 64:
b2 = 36
Take the square root:
b = 6
Therefore, m is either 2 − 6 = −4 or 2 + 6 = 8, and the slope of this line is either 8/6 = 4/3 or 8/(−6) = −4/3. The only equation among the choices that is satisfied by the ordered pair (2, 3) and has a slope of either 4/3 or −4/3 is (B).
11. D Recall that the slopes of perpendicular lines are opposite reciprocals. The slope of the segment from (0, 0) to (5, 6) is 6/5, so the slope of its perpendicular is −5/6.
Therefore
Cross-multiply:
−36 = 5(5 − k)
Distribute:
−36 = 25 − 5k
Subtract 25:
−61 = −5k
Divide by −5:
61/5 = k
12. 12 The slope of this line is 
,
therefore,
Cross-multiply:
3k − 12 = 24
Add 12:
3k = 36
Divide by 3:
k = 12
13. 4/3 or 1.33 Since the slope and y-intercept are given, it is easy to express the linear equation in slope-intercept form: y = 3x − 4.
The x-intercept is the value of x on the line for which y = 0:
0 = 3x − 4
Add 4:
4 = 3x
Divide by 3:
4/3 = x
14. ¼ or .25 The line contains the points (−1, 1) and (3, 2), so its slope is 
15. 5/2 or 2.5 Although we could solve this problem by deriving the linear equation, it is perhaps easier to take advantage of the result from question 14. The slope of 1/4 means that the y-coordinate of any point on the line increases by 1/3 each time the x-coordinate increases by 1. Since the x-coordinate increases by 2 between f(3) and f(5), the y-coordinate must therefore increase by 2(1/4) = 1/2, so f(5) = 2 + ½ = 2.5.
16. 4 The line contains the points (2b, 0) and (0, −8b); therefore, it has a slope of 
.
17. 3/5 or .6
Multiply by 5x:
10 + 2 = 20x
Simplify:
12 = 20x
Divide by 20:
x = 12/20 = 3/5
18. 4 The x-intercept is the value of x for which y = 0:
3x − 2(0) = 12
Simplify:
3x = 12
Divide by 3:
x = 4
19. 15 Recall that the slope of a line in standard form ax + by = c is −a/b. Therefore, the slope of 5x − 2y = 5 is 5/2 and the slope of 6x + ky = 9 is −6/k. If these lines are perpendicular, then their slopes are opposite reciprocals:
Multiply by 6:
k = 30/2 = 15
20. A The total revenue for selling n calculators at $98 each is $98n the cost for making and selling n calculators is $(35n + 120,000). Therefore the profit is $(98n − 35n − 120,000) = 63n − 120,000 dollars.
21. C This line contains the points (5, 0) and (0, 6) and therefore has a slope of 
.
Since its y-intercept is 6, its slope-intercept form is 
or, subtracting 6 from both sides, 
22. B Since f is a linear function, it has a slope that we can call m. Recall that it’s often useful to think of the slope of a line as the “unit change,” that is, the amount that y changes each time x increases by 1. Since the x values increase by 1 with each step in our table, the y values must therefore increase by m with each step. This means that a = 8 − m and b = 8 + m. Therefore, a + b = 8 − m + 8 + m = 16.
On the real number line, a number, b, is more than twice as far from −3 as it is from 3. Which of the following equations can be solved to find all possible values of b?
A) |b − 3| > 2|b + 3|
B) |b + 3| > 2|b − 3|
C) 2|b − 3| > |b + 3|
D) 2|b + 3| > |b − 3|
(Medium-hard) We can use this definition to translate the problem. The key is to translate the statement “b is more than twice as far from −3 as it is from 3” into a statement about distances: “The distance between b and −3 is more than twice the distance between b and 3.” Notice how easily this translates into an inequality:
(The distance between b and −3) is more than (twice the distance between b and 3)
|b − (−3)| > 2|b − 3|
|b + 3| > 2|b − 3|
which is choice (B).
If 
, what is one possible value of x?
(Easy) This kind of inequality is called a “sandwich inequality” because the expression in the middle is between the other two, like meat between slices of bread. Working with inequalities like this one requires understanding the Laws of Inequality.
The Laws of Inequality
Every inequality is a “tipped” scale, and the Laws of Inequality are simply the rules for “keeping the scale tipped the right way,” that is, deducing other true inequalities that follow from the original one. In a nutshell, the Laws of Inequality say that
1. You may make changes to any inequality, as long as you follow rules 2, 3 and 4.
2. Whatever you do to one side of the inequality, you must do to the other.
3. You may not perform undefined operations (like dividing by 0), or operations that have more than one possible result (like taking a square root).
4. If you multiply or divide both sides by a negative number, you must “switch” the direction of the inequality. This is because multiplying or dividing by a negative number involves a reflection over the origin on the number line, and this reflection requires the “switch”:
So we can solve the sandwich inequality by applying the correct laws of inequality:
Multiply by −6 (the common denominator) and “switch:”
3 > 12x − 6 > 2
Add 6:
9 > 12x > 8
Divide by 12:
0.75 > x > 0.66 …
Therefore, any value greater than 0.666 but less than 0.750 is correct.
Which of the following must be true if 
?
A) a ≤ −3b
B) a ≥ −3b
C) a ≤ −3b < 0 or a ≥ −3b > 0
D) a ≤ −3b < 0 or a ≥ −3b > 0
(Hard) We might be tempted to multiply both sides of the inequality by b and get the inequality in (A). But this would be incorrect because it would ignore rule 4. We need to consider the possibility that b might be negative. Let’s think about possible solutions to the original inequality. Notice that a = 10 and b = −2 gives a possible solution, because 10/(−2) = −5 ≤ −3. But this would not satisfy the inequality in (A): 10 is not less than or equal to (−3)(−2) = 6.
To solve this inequality, we will need to consider two distinct possible conditions:
Condition 1: If b > 0, then a ≤ −3b and therefore a ≤ −3b < 0
Condition 2: If b < 0, then a ≥ −3b and therefore a ≥ −3b > 0
which is the answer in choice (D).
On the real number line, a number, b, is more than twice as far from −3 as it is from 3. Which of the following graphs represents all possible values of b?
A)
B)
C)
D)
(Medium) We saw this scenario in Lesson 8, but now we are asked to graph the solution. Recall from Lesson 8 that this relationship is expressed by the inequality |b + 3| > 2|b − 3|. How do we translate this into a graph? The simplest way to start is to visualize the number line, and to think about a related, but simpler, question: What if b is exactly twice as far from −3 as it is from 3? A little guessing and checking should reveal that two points work:
Notice that 9 works because 3 is the midpoint between −3 and 9, and 1 works because it is 2/3 of the way from −3 to 3. Also, you can confirm that both numbers satisfy the equation |b + 3| = 2|b − 3|. These two points now divide the line into three parts: everything less than 1, everything between 1 and 9, and everything greater than 9. A little bit of checking (just pick a number from each portion and plug it into our inequality) confirms that only the numbers in the middle portion satisfy our inequality, so the correct graph is the one in choice (D).
What positive number is twice as far from 10 as it is from 1?
If the points (2, a) and (14, b) are 20 units apart, what is |a − b|?
What is the least integer n for which 
?
If |x + 4| = |x − 5|, what is the value of x?
What is the greatest integer value of n such that 
?
What is the only integer b for which 
and 3b ≥ 7.5?
If (b + 2)2 = (b − 5)2, what is the value of b?
Which of the following statements is equivalent to the statement −4 < 2x ≤ 2?
A) x > −2 and x ≤ 1
B) x < −2 or x ≥ 1
C) x ≥ −2 and x < 1
D) x ≤ −2 or x > 1
The annual profit from the sales of an item is equal to the annual revenue minus the annual cost for that item. The revenue from that item is equal to the number of units sold times the price per unit. If n units of a portable heart monitor were sold in 2012 at a price of $65 each, and the annual cost to produce n units was $(20,000 + 10n), then which of the following statements indicates that the total profit for this heart monitor in 2012 was greater than $500,000?
A) 500,000 < 55n − 20,000
B) 500,000 > 55n − 20,000
C) 500,000 < 55n + 20,000n
D) 500,000 < 75n − 20,000n
Colin can read a maximum of 25 pages an hour. If he has been reading a 250 page book for h hours, where h < 10, and has p pages left to read, which of the following expresses the relationship between p and h?
A) 
B) 
C) 250 − p ≤ 25h
D) 250 + 25h ≤ p
On the real number line, a number, x, is more than 4 times as far from 10 as it is from 40. Which of the following statements describes all possible values of x?
A) x < 34 or x > 50
B) x > 40
C) 34 < x < 50
D) 32.5 < x < 160
If a < 0 and |a − 5| = 7, what is |a|?
If n is a positive integer and 16 > |6 − 3n| > 19, what is the value of n?
What is the only integer n such that 20 − 2n > 5 and 
?
What is the smallest number that is as far from 9.25 as 3 is from −1.5?
If |2x + 1| = 2|k − x|, for all values of x, what is the value of |k|?
Which of the following is equivalent to the statement |x − 2| < 1?
A) x < 3
B) x < −1
C) 1 < x < 3
D) −1 < x < 3
If the average (arithmetic mean) of a and b is greater than the average (arithmetic mean) of c and 2b, which of the following must be true?
A) b > 0
B) a > b
C) a > b + c
D) a + c > b
Of the statements below, which is equivalent to the statement “The distance from x to 1 is greater than the distance from x to 3?
A) 1 < x < 3
B) x > 2
C) x < 2
D) x − 1 > 3
Which of the following is equivalent to the statement 4x2 ≥ 9?
A) 2x > 3
B) x ≥ 1.5 or x ≤ −1.5
C) |x| > 2
D) −1.5 ≤ x ≤ 1.5
The graph above indicates the complete solution set to which of the following statements?
A) |x − 3| > 3
B) |x| < 6
C) |x − 6| < 6
D) |x − 3| < 3
Which of the following is true for all real values of x?
A) |x| > 0
B) x < 2 or x > 1
C) x > −2 or x < −3
D) x2 − 1 > 0
1. 4 It is helpful to plot these values on the number line and think:
The distance between 1 and 10 is 9, so clearly the number that is 9 more units to the left of 1, namely −8, is twice as far from 10 as it is from 1. However, this is a negative number so it can’t be our answer. There is one other number that is twice as far from 10 as it is from 1: the number that is 1/3 the distance from 1 to 10. This number is 4, which is 3 units from 1 and 6 units from 10.
2. 16 From the Distance Formula,
(2 − 14)2 + (a − b)2 = 202
Simplify:
144 + (a − b)2 = 400
Subtract 144:
(a − b)2 = 256
Square root:
|a − b| = 16
3. 8
Since n must be positive for this statement to be true, we can multiply by 9n without having to “swap” the inequality symbols:
0 < 36 < 5n
Divide by 5:
0 < 7.2 < n
Therefore, the smallest integer value of n is 8.
4. ½ or .5 Two numbers, a and b, have the same absolute value only if they are equal or opposites. Clearly x + 4 and x − 5 cannot be equal, since x − 5 is 9 less than x + 4. Therefore they must be opposites.
x + 4 = −(x − 5)
Distribute:
x + 4 = −x + 5
Add x:
2x + 4 = 5
Subtract 4:
2x = 1
Divide by 2:
x = 1/2
5. 10
Multiply by −42 and “swap:”
2n < 21
Divide by 2:
n < 10.5
Therefore, the greatest possible integer value of n is 10.
6. 3
3b ≥ 7.5
Divide by 3:
b ≥ 2.5
Since b is greater than or equal to 2.5, it is positive, so we can multiply both sides by 11b without “swapping” the inequality:
11 > 3b
Divide by 3:
3.67 > b
The only integer between 2.5 and 3.67 is 3.
7. 3/2 or 1.5
(b + 2)2 = (b − 5)2
FOIL:
b2 + 4b + 4 = b2 − 10b + 25
Subtract b2:
4b + 4 = −10b + 25
Add 10b:
14b + 4 = 25
Subtract 4:
14b = 21
Divide by 14:
b = 1.5
8. A
−4 < 2x ≤ 2
Divide by 2:
−2 < x ≤ 1
which is equivalent to −2 < x and x ≤ 1.
9. A The profit is the revenue minus the cost: 65n − (20,000 + 10n) = 55n − 20,000.
10. C If Colin can read a maximum of 25 pages an hour, then in h hours he can read a maximum of 25h pages. If he has p pages left in a 250-page book, he has read 250 − p pages. Since it has taken him h hours to read these 250 − p pages, 250 − p ≤ 25h.
11. C
|x − 10| > 4|x − 40|
It helps to sketch the number line and divide is into three sections: the numbers less than 10, the numbers between 10 and 40, and the numbers greater than 40.
CASE 1: x < 10. It should be clear that all numbers less than 10 are closer to 10 than they are to 40, so this set contains no solutions.
CASE 2: 10 < x ≤ 40. If x is between 10 and 40, x − 10 is positive and x − 40 is negative, so |x − 10| = x − 10 and |x − 40| = −(x − 40).
|x − 10| > 4|x − 40|
Substitute:
x − 10 > −4(x − 40)
Distribute:
x − 10 > −4x + 160
Add 4x:
5x − 10 > 160
Add 10:
5x > 170
Divide by 5:
x > 34
CASE 3: x > 40. If x is greater than 40, then both x − 10 and x − 40 are positive, so |x − 10| = x − 10 and |x − 40| = x − 40.
|x − 10| > 4|x − 40|
Substitute:
x − 10 > 4(x − 40)
Distribute:
x − 10 > 4x − 160
Add 10:
x > 4x − 150
Subtract 4x:
−3x > −150
Divide by −3 and “swap:”
x < 50
So this gives us 40 < x < 50. When we combine this with the solutions from CASE 2, we get 34 < x < 50.
12. 2 If |a − 5| = 7, then either a − 5 = 7 or a − 5 = − 7, so either a = 12 or a = −2. Since a < 0, a must be −2, and |−2| = 2.
13. 8 CASE 1: If 6 − 3n is positive, then
|6 − 3n| = 6 − 3n, so
16 > 6 − 3n > 19
Subtract 6:
10 > −3n > 13
Divide by −3 and “swap:”
−10/3 < n < −13/3
But this contradicts the fact that n is positive.
CASE 2: If 6 − 3n is negative, then
|6 − 3n| = −(6 − 3n), so
16 > −(6 − 3n) > 19
Distribute:
16 > −6 + 3n > 19
Add 6:
22 > 3n > 25
Divide by 3:
7.33 > n > 8.33
And the only integer in this range is n = 8.
14. 7
20 − 2n > 5
Subtract 20:
−2n > −15
Divide by −2 and “swap:”
n < 7.5
Multiply by 3:
2n > 12
Divide by 2:
n > 6
Since n must be an integer between 6 and 7.5, n = 7.
15. 4.75 The distance from 3 to −1.5 is |3−(−1.5)| = 4.5. Therefore the two numbers that are 4.5 away from 9.25 are 9.25 + 4.5 = 13.75 and 9.25 − 4.5 = 4.75.
16. ½ or .5 If the equation is true for all values of x, let’s choose a convenient value for x,
like x = 1.
|2x + 1| = 2|k − x|
Substitute x = 1:
|2(1) + 1| = 2|k − 1|
Simplify:
3 = 2|k − 1|
Divide by 2:
1.5 = |k − 1|
Therefore
±1.5 = k − 1
Add 1:
k = 2.5 or −0.5
Now try x = 0:
|2(0) + 1| = 2|k − 0|
Simplify:
1 = 2|k|
Divide by 2:
0.5 = |k|
Therefore
±0.5 = k
Therefore, k = −0.5 and so |k| = |−0.5| = 0.5.
17. C Recall that the expression |x − 2| means “the distance from x to 2,” so the statement |x − 2| < 1 means “The distance from x to 2 is less than 1.” Therefore, the solution set is all of the numbers that are less than 1 unit away from 2, which are all the numbers between 1 and 3.
18. C
Multiply by 2:
a + b > c + 2b
Subtract b:
a > c + b
19. B The formal translation of this statement is |x − 1| > |x − 3|, which we can solve algebraically by considering three cases: (I) x ≤ 1, (II) 1 < x ≤ 3, and (III) x > 3, but it is probably easier to just graph the number line and notice that the midpoint between 1 and 3, that is, 2, is the point at which the distance to 1 and the distance to 3 are equal. Therefore, the points that are farther from 1 than from 3 are simply the points to the right of this midpoint, or x > 2.
20. B
4x2 ≥ 9
Take square root:
|2x| ≥ 3
If x > 0:
2x ≥ 3
Divide by 2:
x ≥ 1.5
If x < 0:
2x ≤ −3
Divide by 2:
x ≤ −1.5
21. D Notice that the midpoint of the segment shown is 3, and the graph shows all points that are less than 3 units in either direction. Therefore, |x − 3| < 3.
22. B (A) is untrue if x = 0, (C) is untrue for x = −2, and (D) is untrue if x = 0.5. But (B) is true for any real number.
Two high school teachers took their classes on a field trip to a museum. One class spent $154 for admission for 20 students and 3 adults, and the other class spent $188 for admission for 24 students and 4 adults. Which of the following systems of equations could be solved to determine the price of a single student admission, s, and the price of a single adult admission, a, in dollars?
A) a + s = 51
44s + 7a = 342
B) 20s + 3a = 154
24s + 4a = 188
C) 
D) 20 + 24 = s
3 + 4 = a
(Medium) This problem can be described with a two-by-two system of equations, that is, two equations with two unknowns. The two equations come from two facts: one class spent $154 for admission and the other class spent $188 for admission. The cost of 20 student admissions and 3 adult admissions is 20s + 3a, so the first equation is 20s + 3a = 154. Similarly, the equation for the other class is 24s + 4a = 188, so the correct answer is (B).
y = 2x − 3
y = −2x + 17
If the solutions to the two equations above are graphed in the xy-plane, what is the y-coordinate of the point at which the graphs intersect?
(Easy) Since the equations of both lines are given in slope-intercept form, we could graph the two lines in the xy-plane to find their point of intersection.
Therefore, the point (5, 7) gives us the only solution to this system, and so the answer to the original question is 7.
Alternately, (as we will see in Lesson 13) we can just add the corresponding sides of the two equations together to get 2y = 14, which yields y = 7.
16x = 4y + k
For what value of k does the system of equations above have at least one solution?
A) −32
B) −30
C) −24
D) −20
(Medium) This is a two-by-two system of linear equations, and so its solution is the intersection of those two lines. If we convert them to slope-intercept form, we get y = 4x + 6 and y = 4x − k/4, which reveals that these two lines have the same slope. This means that they are either parallel lines or identical lines. Two lines with the same slope can intersect only if they are the same line, and therefore −k/4 = 6 and k = −24.
Let’s go back to the second linear system from Lesson 11. This system can also be solved with a simple application of the Law of Substitution.
y = 2x − 3
y = −2x + 17
1. Substitute for y:
2x − 3 = −2x + 17
2. Add 2x:
4x − 3 = 17
3. Add 3:
4x = 20
4. Divide by 4:
x = 5
5. Plug into either original equation to find y:
y = 2(5) − 3 or −2(5) + 17 = 7
3x + y = 3y + 4
x + 4y = 6
Based on the system of equations above what is the value of the product xy?
(Medium) This system is not quite as tidy as the previous one, but we can still solve it by using the Law of Substitution.
3x + y = 3y + 4
x + 4y = 6
Subtract 4y from second equation to isolate x:
x = −4y + 6
Substitute for x in first equation:
3(−4y + 6) + y = 3y + 4
Simplify left side:
−11y + 18 = 3y + 4
Add 11y and subtract 4:
14 = 14y
Divide by 14:
1 = y
Substitute y = 1 to find x:
x = −4(1) + 6 = 2
Evaluate xy:
xy = (2)(1) = 2
3x + 6y = 18
3x + 4y = 6
Based on the system of equations above, what is the value of y?
(Easy) Although this system can be solved by substitution (try it as an exercise), the setup of these equations suggests a much easier method, known as linear combination. It’s based on a simple idea:
If we apply this rule to our system, notice that we can easily eliminate x from the system by just subtracting the equations:
Divide by 2:
y = 6
3x − y = 20
2x + 4y = 7
Based on the system of equations above, what is the value of x − 5y?
(Medium) This question looks tougher than the previous one, because it’s not just asking for x or y. It seems that the question requires us to solve the system for x and y and then to plug these values into the expression x − 5y and evaluate. We could do that, but there is a much simpler method. Notice that a simple combination gives us the expression the question is asking for.
If 3x + 2y = 72, and y = 3x, what is the value of x?
If 2a − 7b = 10 and 2a + 7b = 2, what is the value of 4a2 − 49b2?
If the lines y = −4x − 3 and y = −3x − b intersect at the point (−1, c), what is the value of b?
If the lines 4x + 5y = 13 and 4y + kx = 2 are parallel, what is the value of k?
If the lines 4x + 5y = 13 and 6y − kx = 6 are perpendicular, what is the value of k?
Based on the system of equations above, what is the value of 
?
If ab = −4 and abc = 12, what is the value of 
?
If a and b are constants and the graphs of the lines 2x − 3y = 8 and ax + by = 2 are perpendicular, then what is the value of 
?
5x − y = 11
2x − 2y = 9
Based on the system of equations above, what is the value of 3x + y?
A) −2
B) 0
C) 2
D) 4
Two numbers have a difference of 4 and a sum of −7. What is their product?
A) −33
B) −10.25
C) 8.25
D) 10.25
It costs Emma p dollars to make each of her custom bracelets, which she sells for m dollars apiece. She makes a profit of $60 if she makes and sells 5 of these bracelets, but she only makes a profit of $10 if she makes 5 bracelets but only sells 4 of them. How much does it cost Emma to make each bracelet?
A) $36
B) $38
C) $48
D) $50
If 2y = x + 1 and 4x + 6y = 0, then y =
If 
and 
, then y =
2x − 5y = 20
10x − 25y = 4k
For what value of k does the system of equations above have at least one solution?
At the beginning of the week, the ratio of cats to dogs at Glenna’s Pet Store was 4 to 5. By the end of the week, the number of cats had doubled, while the number of dogs had increased by 12. If the ratio of cats to dogs at the end of the week was 1 to 1, how many cats did the store have at the beginning of the week?
Jenny originally had twice as many friendship bracelets as Emilie. After Jenny gave Emilie 5 of her friendship bracelets, Jenny still had 10 more than Emilie. How many friendship bracelets did Jenny have originally?
The average (arithmetic mean) of x and y is 14. If the value of x is doubled and the value of y is tripled, the average (arithmetic mean) of the two numbers remains the same. What is the value of x?
7m + 10n = 7
6m + 9n = 1
Based on the system of equations above, what is the value of 4m + 4n?
In the xy-plane, perpendicular lines a and b intersect at the point (2, 2). If line a contains the point (7, 1), which of the following points is on line b?
A) (0, 1)
B) (4, 5)
C) (7, 3)
D) (3, 7)
Which of the following pairs of equations has no solution in common?
A) 2x − 3y = 1 and 6x − 9y = 3
B) y = 4x and y = −4x
C) 2x − 3y = 1 and 6x − 9y = 2
D) y = 4x and 2y − 8x = 0
In the xy-plane, the line l is perpendicular to the line described by the equation 
. What is the slope of line l?
A) −2
B) 
C) 
D) 2
1. 8
3x + 2y = 72
Substitute y = 3x:
3x + 2(3x) = 72
Simplify:
9x = 72
Divide by 9:
x = 8
2. 20
4a2 − 49b2
Factor:
(2a − 7b)(2a + 7b)
Substitute:
(10)(2) = 20
3. 2
y = −4x − 3
Substitute x = −1, y = c:
c = −4(−1) − 3
Simplify:
c = 1
Other equation:
y = −3x − b
Substitute x = −1, y = 1:
1 = −3(−1) − b
Simplify:
1 = 3 − b
Subtract 3:
−2 = −b
Divide by −1:
2 = b
4. 3.2 or 16/5 Parallel lines must have equal slopes. The slope of 4x + 5y = 13 is −4/5, and the slope of 4y + kx = 2 is −k/4.
Cross-multiply:
−5k = −16
Divide by −5:
k = 16/5 = 3.2
5. 7.5 or 15/2 Perpendicular line have slopes that are opposite reciprocals. The slope of 4x + 5y = 13 is −4/5, and the slope of 6y − kx = 6 is k/6.
Cross-multiply:
−4k = −30
Divide by −4:
k = 7.5
6. .25 or ¼ First equation:
Divide by 2:
Second equation:
Subtract 1:
Reciprocate:
Multiply:
7. .75 or ¾
abc = 12
Substitute ab = −4:
(−4)c = 12
Divide by −4:
c = −3
Expression to evaluate:
Substitute c = −3 and ab = −4:
8. 4.5 or 9/2 The slope of 2x − 3y = 8 is 2/3, and the slope of ax + by = 2 is −a/b. If the two lines are perpendicular, then the slopes are
opposite reciprocals:
Reciprocate:
Multiply by 3:
9. C
5x − y = 11
2x − 2y = 9
Subtract equations:
3x + y = 2
10. C
a − b = 4
a + b = −7
Add equations:
2a = −3
Divide by 2:
a = −1.5
Substitute a = −1.5:
−1.5 + b = −7
Add 1.5:
b = −7 + 1.5 = −5.5
Evaluate product:
ab = (−1.5)(−5.5) = 8.25
11. B Let c = the cost to make each one of Emma’s bracelets.
5m − 5c = 60
4m − 5c = 10
Subtract:
m = 50
Substitute m = 50
5(50) − 5c = 60
Simplify:
250 − 5c = 60
Subtract 250:
−5c = −190
Divide by −5:
c = 38
12. 2/7 or .286 or .285
2y = x + 1
Subtract 1:
2y − 1 = x
Given:
4x + 6y = 0
Substitute x = 2y − 1:
4(2y − 1) + 6y = 0
Distribute:
8y − 4 + 6y = 0
14y − 4 = 0
Add 4:
14y = 4
Divide by 14:
y = 4/14= 2/7
13. 1/6 or .166 or .167
Add equations:
12x = 2
Divide by 12:
x = 2/12 = 1/6
14. 25 The slope of 2x − 5y = 20 is 2/5. The slope of 10x − 25y = 4k is 10/25 = 2/5. Since the two lines have the same slope, they have no points of intersection unless they are the same line.
2x − 5y = 20
10x − 25y = 4k
Multiply first equation by 5:
10x − 25y = 100
Therefore, 4k = 100 and so k = 25.
15. 16 If the original ratio of cats to dogs is 4 to 5, then we can say there were 4n cats and 5n dogs to start. At the end of the week, therefore, there were 8n cats and 5n + 12 dogs. If this ratio was 1:1, then
8n = 5n + 12
Subtract 5n:
3n = 12
Divide by 3:
n = 4
Therefore, there were 4n = 4(4) = 16 cats at the beginning of the week.
16. 40 Let x = the number of friendship bracelets Emilie had to start. This means that Jenny originally had 2x bracelets. After Jenny gave 5 of them to Emilie, Jenny had 2x − 5 and Emilie had x + 5. If Jenny still had 10 more than Emilie, then
2x − 5 = 10 + (x + 5)
Simplify:
2x − 5 = x + 15
Subtract x and add 5:
x = 20
This means that Jenny had 2x = 2(20) = 40 to start.
17. 56
Multiply by 2:
x + y = 28
If x is doubled and y is tripled, the average remains the same:
Multiply by 2:
2x + 3y = 28
Previous equation:
x + y = 28
Multiply by 3:
3x + 3y = 84
Other equation:
2x + 3y = 28
Subtract equations:
x = 56
18. 24
7m + 10n = 7
6m + 9n = 1
Subtract equations:
m + n = 6
Multiply by 4:
4m + 4n = 24
19. D Line a contains the points (2, 2) and (7, 1); therefore, it has a slope of 
. If line b is perpendicular to line a, then it must have a slope of 5 (the opposite reciprocal of −1/5). You might find it helpful to sketch the line with slope 5 through the point (2, 2), and confirm that is passes through the point (3, 7), which is one unit to the right and one 5 units up.
20. C In order for two lines in the xy-plane to have no points in common, they must be parallel and nonidentical. The only two such lines among these choices are 2x − 3y = 1 and 6x − 9y = 2, which both have a slope of 2/3, but have different y-intercepts of −1/3 and − 2/9.
21. A
Multiply by 2xy:
2y + x = 2x
Subtract x:
2y = x
Divide by 2:
This line has a slope of 1/2, so the perpendicular must have a slope of −2.