Factor:
x2 – 11x + 30 = 0
51
Answer: (x – 5)(x – 6) = 0
Since the last sign is positive, set up 2 parentheses with the sign of the middle term.
(x – )(x – )
Find two numbers that multiply to 30 and add to 11 and place them in the parentheses.
(x – 5)(x – 6)
What values for x solve the equation?
For each of the following, could the answer be an integer if x is an integer greater than 1?
(A) x10 + x−10=
(B) x1/6 + x1/2 =
52
Answer: (A) No; (B) Yes
(A) No. x−10 = 1/x10. For any x > 1, this won't be an integer.
(B) Yes. This is equivalent to 
, so if x has an integer sixth root this will be an integer. For example, if x equals 64, the sixth root of x is 2, and the square root is 8.
Any number with an integer sixth root will have an integer square root. Why?
Is it possible to solve for a single value of x in each of the following systems of equations?
| (A) 2x + 3y = 8 2x − y = 0 |
(B) x2 + y − 24 = 0 y = 2x |
| (C) 2x − 4y = 13 −6x + 12y = −39 |
53
Answer: (A) Yes; (B) No; (C) No
(A) Yes. We are given 2 linear equations. There are no xy terms or x/y terms.
(B) No. There is an x2 term. If you substitute 2x in for y, you can factor the equation to (x + 6)(x – 4) = 0, which produces two possible values for x (namely, 6 and 4).
(C) No. The two equations are equivalent. The second equation is just the first equation multiplied by −3.
What is S25 in the following sequence?
Sn = Sn−1 – 10 and S3 = 0.
54
Answer: −220
First, we need to convert the recursive sequence definition provided into a direct sequence formula. Each term is 10 less than the previous one. Therefore Sn = −10n + k, where k is some constant that we must determine. Use S3 to find a value for k: 0 = −10(3) + k. Thus, k = 30, so Sn = −10n + 30. Now we plug in 25 for n: S25 = −10(25) + 30 = −220.
Alternatively, we could plug in 0 for S3 and find that S4 = −10, S5 = −20, S6 = −30, etc. This pattern is that the subscript (4, 5, 6) is 3 more than the tens digit (1, 2, 3). Thus, S25 = −220.
If −5 is one solution to the equation x2 + kx – 10 = 0, where k is a constant, what is the other solution?
55
Answer: x = 2
If one solution is –5, we know one of the factors of the quadratic expression is (x + 5). We now know the other factor is (x – 2) because the two numbers in parentheses must multiply to −10. Therefore the other solution is x = 2.
Bonus: What is k?
From the solution above, if (x + 5)(x – 2) = 0, then distributing yields x2 + 3x – 10 = 0. Thus, k = 3.
Alternatively, plug x = −5 into the original quadratic and solve for k. (−5)2 + k(−5)−10 = 0 becomes 25 – 5k – 10 = 0, or 15 = 5k. Again, k = 3.
If ax < ay, is x < y?
56
Answer: Maybe (it depends on the sign of a)
It may be tempting to simply divide by a on both sides. However, this will only yield x < y if a is positive. Remember that when you multiply or divide an inequality by a negative number, you must flip the sign. So if a is negative, the conclusion would be that x > y. Depending on the sign of a, the answer could be yes or no.
Solve for y:
y2 + 7y – 60 = 0
57
Answer: y = −12, 5
Since the last sign is negative, set up 2 parentheses with opposite signs. (y + )(y – )
Find two numbers that multiply to 60 and differ by:
12 × 5 = 60 12 – 5 = 7
Place the larger number in the parentheses with the same sign as the middle term (+7y):
(y + 12)(y – 5) = 0
If y + 12 = 0, then y = −12. If y – 5 = 0, then y = 5.
What is the value of x?
53x = 57x – 4
58
Answer: 1
Since the bases are equal, we can simply set the exponents equal to each other.
3x = 7x − 4
4 = 4x
1 = x
What is the minimum value of f(x) = −5 + (x + 7)2, and at what value of x does it occur?
59
Answer: minimum value = −5, x = −7
The squared expression will always be non-negative, so to make f(x) as small as possible, make the squared expression as small as possible—set it equal to zero. If x + 7 = 0, x = −7. Once you have the x value, plug it back into the original equation to solve for the minimum value. f(x) = −5 + (0)2. Therefore, the minimum value is −5.
Remember, f(x) and y are synonymous.
What are all possible values of x?
x2 – 27x + 50 = 0
60
Answer: x = 2 or 25
Since the last sign is positive, set up 2 parentheses with the sign of the middle term.
(x – ) (x – )
Find two numbers that multiply to 50 and add to 27 and place them in the parentheses.
(x – 2) (x – 25) = 0.
If x – 2 = 0, then x = 2. If x – 25 = 0, then x = 25.
Solve for b:
61
Answer: b ≤ −28
To isolate b, multiply both sides by −7 and flip the direction of the inequality sign.
When multiplying or dividing an inequality by a negative number, remember to switch the direction of the inequality sign.
Use factoring to simplify the following expressions:
(A) 45 + 45 + 45 + 45
(B) xw + yw + zx + zy
62
Answer: (A) 46; (B) (w + z)(x + y)
(A) The greatest common factor is 45.
45(1 + 1 + 1 + 1) = 45(4) = 46.
Make sure to look for common terms that can be factored out of an expression. Factoring is often a crucial step toward solving an equation.
(B) Factor by grouping: (xw + yw) + (zx + zy) =
w(x + y) + z(x + y) = (w + z)(x + y).
If you have 4 terms and 4 variables, look to factor by grouping.
Solve for each of the following:
(A) If 
, what is 2x + y?
(B) If 
, what is 3r + 6t?
63
Answer: (A) 7; (B) 75
(A) Multiply both sides by 2 and add y to each side.
(B) Square both sides and multiply by 3.
Distribute:
(b + 7)(b – 10)
64
Answer: b2 − 3b − 70
Use FOIL—First, Outer, Inner, Last
(b)(b) + (b)(−10) + (7)(b) + (7)(−10)
b2 – 10b + 7b – 70
b2 – 3b – 70
If 2 is one solution to the equation x2 – 9x + c = 0, where c is a constant, what is the other solution?
65
Answer: 7
Work backwards—even though we do not know the value of c, since 2 is one solution, we know the factored form of the quadratic is (x – 2)(x − ?). We also know that the two numbers in parentheses must add to −9. Therefore the factored form is (x – 2)(x – 7) and the other solution is x = 7.
This problem can also be solved by plugging x = 2 into the original equation and solving for c, then factoring the resulting equation (x2 – 9x + 14 = 0).
What error has been made?
x2 = 36
x = 6
66
Answer:
Remember, 
. So after we take the square root of both sides, we have |x| = 6.
This gives two possibilities: x = 6 or x = −6.
Alternatively, simply recall that there are generally two possible solutions in exponential equations with an even exponent. Thus when x2 = 36, x = 6 or −6.
If c < 4, what is the range of possible values of d for the equation 3c = −6d?
67
Answer: d > −2
If we isolate c in the equation, we can then substitute into the inequality to find the range of d.
3c = −6d
c = −2d
Now replace (c) with (−2d) in the inequality:
(−2d) < 4
d > −2
What are the roots of x3 – x = 0?
68
Answer: x = 0, −1, or 1
Factor the equation, since we already have 0 on one side:
x(x2 – 1) = 0
x(x + 1)(x – 1) = 0
x = 0, −1, or 1.
The temptation is to move x to the other side and divide both sides by x, leaving us with x2 = 1. Avoid dividing away a variable unless you know it does not equal 0.
Consider the formula 
If a is doubled and b is increased by a factor of 4, by what factor is H increased?
69
Answer: H is increased by a factor of 2 (H is doubled)
The exponent of 3 on a means when we double a, the whole formula gets multiplied by 23, or 8. b has no exponent, but it is in the denominator, so quadrupling it is the equivalent of multiplying the formula by 1/4. Thus, H gets multiplied by 8 × 1/4 = 2.
What is x? (Hint: Try a method other than substitution)
x + y = 10
3x – 5y = 6
70
Answer: 7
One way to solve for a variable when you have two equations is to combine the equations in a way that eliminates the other variable, (here, y). In this case, we can multiply the first equation by 5, and then add it to the second equation, giving us:
On the GRE, combination is often faster than substitution.
If x and y are both less than 5, what is the maximum value for the product xy?
71
Answer: Positive infinity (i.e. there is no maximum value)
Example: (−10)(−4) = 40
(−200)(−32) = 6,400
(−500,000)(−11,000) = 5,500,000,000
Don't forget about negative numbers!
Solve for w:
22w = 8w – 5
72
Answer: w = 15
We must first obtain the same base on both sides. Convert the 8 into a power of 2:
22w = (23)w – 5 22w = 23w – 15
Now that the bases are equal, we can set the exponents equal to each other:
2w = 3w – 15 
w = 15.
Solve:
(x – 4)2 = 49
73
Answer: x = 11 or −3
Do not multiply out (x – 4)2 if there is a perfect square on one side of the equation. Instead, take the square root of both sides, and remember to place the side of the equation containing the unknown in an absolute value. |x – 4| = 7. Our two solutions to this equation are x – 4 = 7 and x – 4 = −7. Solving these two equations gives us x = 11 and −3.
The first few steps of a problem are shown. Finish the problem and answer the question: what is x?
74
Answer: x = 6 (x does NOT equal 1!)
Although this equation can be simplified and factored into (x − 6)(x − 1) = 0, you need to be careful. When you square an equation containing a variable, you may create extraneous solutions. Potential answers need to be plugged back in to the original equation and verified. 6 is a genuine solution; 1 is not.
Try plugging 1 back into the original equation to verify that x cannot equal 1.
The square root symbol always indicates the positive root.
What is x + y + z?
x + y = 8
x + z = 11
y + z = 7
75
Answer: 13
There is often a faster method than solving for the value of each variable. In this case, we can simply add all the equations together!
Remember, x + y + z is a “combo.” In this type of problem there is a good chance you will not need to determine the individual values of the variables.
Simplify: 
76
Answer: −7
Remember, (a + b)(a – b) = a2 – b2.
Therefore, our expression is equal to:
(2 – 9) × (4 – 3) = (−7)(1) = −7
A group of rabbits grows by a constant factor every day. If the population grew from 200 to 5,000 in one week, by what factor does its population increase every day?
77
Answer: 
The number of rabbits at the beginning of the day times some factor x equals the new number of rabbits. After a week the original value will have been multiplied by x7:
If a ≠ b, a > 0 and b > 0, simplify:
78
Answer: 
When there is a square root term in the denominator that is added to or subtracted from another term we can multiply by the conjugate (the same expression, but with the sign on the 2nd term flipped) to simplify:
Alternatively, you could use the special product a2 – b2 = (a + b)(a – b) to solve. In this case, 
, and so the term 
would cancel from the top and bottom, leaving 
Solve for x:
79
Answer: x = 24
Divide the second equation by 2 and get 3y + 5z = 9. Substitute 9 for the denominator of the fraction in the first equation. This gives 
, which reduces to 
, and finally x = 24.
Remember, when you see 3 variables and only 2 equations, you should not automatically assume that you cannot solve for a particular value.
Simplify:
80
Answer: 
To remove a square root from a denominator of the form 
, multiply the fraction by 
. The form is the same whether you are dealing with numbers, variables, or a combination of the two.
In a sequence of terms, each term after the first is 23 times the previous term. If the 19th term is 40, what is the 11th term?
81
Answer: 
We could simply work backwards from the 19th term, dividing each term by 23. S18 = 40/23, S17 = 40/232, etc.
Generally, in a sequence, if you know the factor that each term is being multiplied by 23 in this case, and if you know just one term, it is sufficient to solve for any other term in the sequence.
Look for a pattern. Each term before the 19th is 40 divided by some power of 23. The power of 23 is determined by how many terms prior to the 19th the term is. For example, S16 would be 
because it is 19 – 16 = 3 terms prior to 19. Since S11 is 19 – 11 = 8 terms prior to 19, S11 = .
How would you factor each of the following expressions?
(A) x5 – x3
(B) 48 + 49 + 410
(C) mn − 2 – 3mn + 4mn + 1
82
Answer:
(A) The GCF is x3, the smaller power.
x3(x2 – 1) = x3 (x + 1)(x – 1).
(B) The GCF is 48. 48(1 + 41 + 42) = 48(21).
(C) The smallest power of m is the GCF. Here it is mn − 2:
mn − 2(1 – 3m2 + 4m3).
Solve by picking numbers:
Bottle 1, with capacity x liters, is half full of water. Bottle 2, with capacity y liters, is one sixth full. If Bottle 2 is three times the size of Bottle 1 and the contents of Bottle 1 are emptied into Bottle 2, how many liters of water, in terms of y, are in Bottle 2?
(A) 
(B) 
(C) 
(D) 
(E) 
83
Answer: (D) 
When problems involve many fractions and no specific quantities, pick numbers that are multiples of all the denominators in the problem. The least common multiple of 6 and 2 is 6. Thus, let the capacity of Bottle 1 = 6 and the capacity of Bottle 2 = 18. Bottle 1 holds 3 liters and bottle 2 holds 3 liters. Bottle 1 is dumped into Bottle 2, which then contains 6 liters. Test each answer choice with y = 18 and notice that (D) is the solution, since 
Set up an appropriate equation to describe the given scenario:
The elasticity (e) of a material is directly proportional to the square of its density (d) and inversely proportional to the cube of its mass (m).
84
Answer: 
A constant k is used in expressions of direct or inverse proportionality. e is directly proportional to d2, which means e = kd2. e is also inversely proportional to m3, so e = k/m3. Putting these two equations together, we get .
Note that k in the final equation must be the product of the k constants in the first two equations, but since k could be any value, we can repeat the use of k for simplicity.
If x + y = 5 and x2 – y2 = 20, what is y?
85
Answer: y = 0.5
First, factor x2 – y2:
(x + y)(x – y ) = 20
Since (x + y) = 5, (x – y ) = 4.
Add these two linear equations, to cancel y: 2x = 9, so x = 4.5
Plugging into x + y = 5, if x = 4.5, then y = 0.5.
Identify the error:
8!+2 ≤ x ≤ 8!+10 implies that 2 ≤ x ≤ 10.
86
Answer: 2 ≤ x – 8! ≤ 10 is the correct result.
In a compound inequality, you must perform the same operation to all 3 expressions, not just the outside expressions. If you subtract 8! from all 3 expressions, you get 2 ≤ x – 8! ≤ 10.
If xz > 0 and yz < 0, what can you conclude about xy?
87
Answer: xy < 0
xz > 0 means x and z have the same sign. yz < 0 means y and z have opposite signs. Together, this means that x and y must have opposite signs and consequently xy < 0.
Factor:
88
Answer: 
This expression is a slightly more complicated version of the special product a2 – b2 = (a + b)(a – b). Notice that x2, 9, 25, and y2 are all perfect squares:
The first three terms of a linear sequence are −2, 18, and 38. What is the rule for this sequence?
89
Answer: Sn = 20n – 22, for integer n > 0
Since the terms are increasing by 20, we know the rule is Sn = 20n + k. Use any of the three given terms to solve for k:
S2 = 20(2) + k 18 = 40 + k k = −22
The rule is Sn = 20n – 22.
Try solving for k using S1 or S3 and verify that you get the same value. Could you express this sequence using a recursive definition?
If 10 ≤ m ≤ 20 and −2 ≤ p ≤ 15 and m and p are both integers, what is the maximum possible value for m – p?
90
Answer: 22
To maximize m – p, make m as large as possible and make p as small as possible. m = 20 and p = −2.
20 – (−2) = 22.
Solve by picking numbers for x and y, then testing the answer choices.
What is the average of (x + y)2 and (x – y)2?
(A) 2x2 – 2y2
(B) x2 + 4xy + y2
(C) x2 + y2
91
Answer: (C) x2 + y2
Let's pick x = 3 and y = 2.
(3 + 2)2 = 25 and (3 – 2)2 = 1, so the average is 
Now, let's test each answer choice:
A) 2(3)2 – 2(2)2 = 10
B) (3)2 + 4(3)(2) + (2)2 = 37
C) (3)2 + (2)2 = 13
Alternatively, expand the expressions:
(x + y)2 = x2 + 2xy + y2
Everyone in a certain office orders a cup of coffee. The ratio of cappuccinos to lattes to espressos ordered is 1 : 2 : 3. If there are 60 people in the office, how many cups of each type of coffee were ordered?
92
Answer: 10 cappuccinos, 20 lattes, 30 espressos
Using the unknown multiplier, we can set up the equation 1x + 2x + 3x = 60, since 60 cups of coffee were ordered (one for each person). Solving for x, we find that x = 10, and then we can apply that multiple to each element in the proportion.
The average of 177, 176, 189 and x is 180. What is x?
93
Answer: 178
A shortcut to dealing with averages is to focus on how much above or below the average every number in the set is. Then, we can “balance” this difference in the final term. In this example, 177 is 3 below the average, 176 is 4 below the average, and 189 is 9 above the average.
−3 – 4 + 9 = +2, so the fourth number must be 2 below the average to balance it out. 178 is 2 below 180.
This method is much easier than applying the definition of averages:
Ugh.
Which is the correct expression?
The phone call pricing for a long distance company is as follows: $5.00 for the first minute, and $0.15 for every additional minute. After 10 minutes, the price drops to $0.10 per minute. How much does a 17 minute phone call cost, in dollars?
(A) 5 + 10(0.15) + 7(0.10)
(B) 5(10) + 0.15 + 7(0.10)
(C) 5 + 9(0.15) + 7(0.10)
94
Answer: (C) 5 + 9(0.15) + 7(0.10)
We know that the first minute costs $5.00. The next 9 minutes (not 10—don't forget to subtract out the first minute!) will be charged at the rate of $0.15 per minute. After that, the next 7 minutes will be charged at the rate of $0.10 per minute.
In a round of miniature golf, the probability that Jasper will get a hole in one is 1/13. The probability that Cornelius will get a hole in one is 1/12. If these probabilities are independent, what is the probability that neither of them will get a hole in one on the next hole?
95
Answer: 11/13
If Jasper has a 1/13 chance of getting a hole in one, that means he has a 12/13 chance of not getting it. Similarly, Cornelius has an 11/12 chance of not getting the hole in one. Because we want the probability of Jasper not getting the hole in one AND Cornelius not getting the hole in one, we multiply the probabilities:
This equation only holds if the two occurrences are independent (like successive coin flips), but many, if not most, probability problems involving more than one event assume independence.
Which of the following changes to a set of at least 3 consecutive positive integers will result in a list whose median and mean are different?
(A) Every number in the list is tripled
(B) Every number in the list has 12 added to it
(C) Every number in the list is squared
96
Answer: (C) Every number in the list is squared
As long as a set of numbers is evenly spaced, its average will equal its median. The changes described in answer choices (A) and (B) would keep the numbers in the lists equally spaced. Only answer choice (C) would change the spacing.
1, 2, 3 3, 6, 9 evenly spaced
1, 2, 3 13, 14, 15 evenly spaced
1, 2, 3 1, 4, 9 not evenly spaced
Initially, the ratio of potbellied pigs to carrots in a room had been 1 : 60. After the potbellied pigs ate most of the carrots, though, the new ratio was 3 : 1. If there were 6 potbellied pigs in the room, how many carrots did they eat, total?
97
Answer: 358 carrots
From the first ratio, we know that there were originally 360 carrots, since 1/60 = 6/360. The second ratio tells us that the potbellied pigs only left 2 carrots uneaten, since 3/1 = 6/2. We can calculate that 360 carrots – 2 uneaten carrots = 358 carrots.
Sean is 15 years older than Eric. In 6 years Sean will be twice as old as Eric. How old is Eric?
(A) 9
(B) 14
(C) 24
98
Answer: (A) 9
An alternative approach to setting up equations and solving for the appropriate variable is to use the answer choices to help you. To demonstrate using the right answer, we begin by assuming that Eric is 9. From the first sentence, we know that Sean is 24. In 6 years, Eric will be 15 and Sean will be 30. 30 is indeed 2 times 15, so we know that 9 is the right answer.
For this type of question, using the answer choices is an alternate approach to setting up the equations. To do very well on the GRE, it's important to know how to setup and solve the algebra, but you should also have other tools available to you.
Give an example of a set of numbers in which 75% of the numbers are greater than or equal to the median.
99
Answer: Can vary, but any set like {1, 2 | 3, 3 | 3, 3 | 5, 9} or {50 | 51 | 51 | 109} {1, 2, 3 | 5, 5, 5 | 5, 6, 7 | 8, 9, 10}. The | mark separates quartiles.
In order for 75% of the terms to be a whole number of terms, the number of terms in the set must be a multiple of 4. So, the number of terms in the set must be even, and the median is the average of the two middle terms.
A key phrase is “…or equal to the median.” By definition, the median is greater than or equal to 50% of the terms, and less than or equal to 50% of the terms. In order for 75% of the terms to be greater than or equal to the median, the terms in the 2nd quartile must equal the median.
Jeff completed a 40 mile circuit around the city, jogging at a constant speed of 5 mph. Then Jeff completed the same circuit again, but this time he biked at a constant speed of 20 mph. What was Jeff's average speed for the total trip?
100
Answer: 8 mph
To calculate average speed, you need the total distance traveled and the total time spent traveling. The distance is straightforward. Jeff traveled 40 miles twice, for a total distance of 80 miles. If we divide distance by speed, we find that it took him 40/5 = 8 hours to jog the circuit, while it took him 40/20 = 2 hours on the bike, for a total time of 10 hours.
Finally, the average speed is the total distance divided by the total time:
