Linear equations and inequalities
A constant theme in the study of mathematics is to take things back to basics. Linear equations and inequalities are the basic building blocks for the solution of all equations in mathematics.
Simple linear equations
All simple linear equations take the form ax + b = c, and the solution to this equation is 
. The trick, of course, is to get the complicated “simple” linear equation into this basic form. The guiding principle is to gather common terms—those involving the variable in question on one side of the equation, and all other terms on the other side of the equation.
(Note: If you were expecting a different answer because of the direction to multiply, remember that part of the process of multiplying fractions is to cancel where possible.)
Be warned: Not all answers will be “nice” integers.
There are many applications involving linear equations. Most involve systems of equations and will be looked at later in this chapter. The next two example problems are meant to highlight the importance of clearly defining the variable for an application and using the units of the problem to write an equation.
Solve each of the following equations for the variable in the problem.
1. 4(2a − 3) − 3(8 − 2a) = −78
2. 5y + 19 + 2(3 − 4y) = 7y − 12
3. 
4. 
5. Solve for a: 3(2a − 5t) − 4(3a + 2t) = 8a − 25t
6. 
7. Kristen has categorized the songs on her MP3 player as rock (defined as any song her parents would not listen to) and classical (songs her parents would listen to). Kristen has 3670 songs on her MP3 player, and the number of rock songs is 110 more than seven times the number of classical songs. How many songs of each kind does she have on her MP3 player?
8. The garden department at Home Station is having a spring sale on plants. Diane bought a total of 90 plants for a total cost (before tax) of $455. Diane only bought plants that were on sale either for $4.50 each or for $6.50 each. How many plants of each kind did she buy?
Linear inequalities
The most challenging thing to remember when solving simple linear inequalities is to reverse the orientation of the inequality when both sides of the sentence are multiplied or divided by a negative number.
Examine the set of numbers graphed on the accompanying number line.
The set contains all the points from −2, which is included in the set, through 5, which is not included. That is, using x as the variable of the inequality, x ≥ −2 and x = 5. This is usually written in the more condensed form −2 ≤ x = 5 (x is between −2, included, and 5, excluded). This is an example of a compound inequality.
Examine the set of numbers graphed on the accompanying number line.
The set of numbers shows all those numbers that are less than or equal to −3 OR those numbers greater than 2. Written in mathematical notation, x ≤ −3 or x > 2. It is important for you to realize that there is no other way to write this equality.
Solve each of the following inequalities and graph the solution on a number line.
1. 8 + 3x > 5x – 4
2. 2(3x − 5) − 3(7 − 4x) ≥ 15x + 14
3. 17 ≤ 3x − 10 = 29
4. −8 < 12 − 5x ≤ 17
5. 3x − 2 = 12 or 5x − 8 ≥ 17
6. 4 − 5x < 19 or 3x + 10 = 25
System of linear equations—graphical
Determining the values of the variables that make multiple equations true at the same time is important because most applications of mathematics involve the issue of meeting multiple requirements simultaneously. For example, business people want to know the point at which the money they spend to put products on the market—their cost—will be gained back from the money taken in by sales—their revenue. The point at which cost = revenue is called the breakeven point.
Systems of equations can be solved graphically as well as algebraically. In this section, you will study graphical solutions.
Please beware: When using graphing utilities, the window dimensions may need to be changed so that the point of intersection is visible on the screen.
System of linear equations—substitution
The substitution method for solving systems of equations is best applied when at least one of the equations in the system is of the y = form.
System of linear equations—elimination
Although the substitution method can be used when the equations in the system are in standard form (Ax + By = C), the process is cumbersome and offers too many opportunities to make a mistake. The elimination (or multiplication−addition) method is a better choice. The goal in this method is to get the coefficients of one of the variables to be equal in size and opposite in sign.
Please note: You should always check your answer in the original equation to be sure you didn’t make a mistake in an intermediate step of the solution.
System of linear equations—three variables
Equations in three variables can be graphed in a three-dimensional system—not something most classrooms have at their disposal. Equations in more than three variables do not have a physical representation available but they do represent the ability for the users of mathematics to think in abstract terms.
In this section, you will learn to solve systems of three linear equations in three variables using the elimination method. This method can be extended to any number of equations (having the same number of variables) to find a solution (if a solution exists). The process is to take one of the equations and pair it against the remaining equations. The same variable will be eliminated from each of these pairs, creating a new system of equations with one less equation and one less variable.
System of linear equations—matrix
In the traditional approach to solving the algebraic equation ax = b, you divide both sides of the equation by a to get the solution 
Because there is no operation called division in matrix algebra, the solution to the equation AX = B is X = A−1 B, where A−1 is the inverse of matrix A. The matrix approach works best with a calculator that has matrix capabilities. While a matrix solution can be obtained using a pencil and paper approach, it will usually be much more cumbersome than the approaches shown in the earlier sections of this chapter.
The matrix equivalent to the system of equations
has the coefficient matrix 
variable matrix 
and matrix of constants 
The solution is 
or (11, 15) when written as an ordered pair.
The matrix solution to the system of equations
has coefficient matrix 
variable matrix 
and matrix of constants 
The solution is 
or the ordered triple (4, 7, −5).
A big advantage to solving systems of linear equations with a matrix solution is that the same amount of work is needed to solve a system with four equations in four variables as is needed to solve a system of equations with two equations in two variables.
System of linear equations—application
There are many applications of mathematics which can be solved with systems of linear equations. As you learned in Algebra I, it is to your advantage to clearly define the variables used to write the equations.
Write systems of equations for each of the following problems. Solve the systems using the elimination method or a matrix equation.
1. Tickets for the fall drama production at Eastside High School were sold at three levels: student tickets purchased in advance, student tickets purchased on the day of the performance, and adult tickets (no matter when the tickets were purchased). There were three performances of the show: Friday night, Saturday night, and a Sunday matinee. The financial report shows the following results for ticket sales: Friday’s show had 150 student advance tickets, 75 student tickets sold at the door, and 300 adult tickets; Saturday’s show had 200 student advance tickets, 100 student tickets sold at the door, and 350 adult tickets; Sunday’s show had 50 student advanced tickets, 70 student tickets sold at the door, and 250 adult tickets. Ticket receipts for the three nights were: Friday $6300, Saturday $7650, and Sunday $4755. What was the price charged for each type of ticket?
2. Tickets for the spring musical at Bayview High School were being sold to students for $8 in advance of the performance, $10 on the day of the performance, and $12 for adults (no matter when the ticket was purchased). The financial report after the musical performances showed that 1250 tickets were sold with receipts totaling $13,400. The number of adult tickets sold exceeded the total number of student tickets sold by 150. How many tickets of each type were sold for the musical?
3. A newer version of the tiled word game has different values for each of the letters. Although underlined letters still score double their value, letters in bold font each triple their value. Determine the values for the letters B, A, E, and R if BARE is worth 111points, BEAR is worth 113 points, RABE is worth 97 points, and BRAE is worth 99 points.
System of linear inequalities
When graphing inequalities on a number line, the difference in the graphs of x > 1 and x ≥ 1 is to use an open circle at 1 for x > 1 (indicating that the endpoint is not included— this is called a half line) and a closed circle for x ≥ 1 (indicating that the point is included—a ray). In the number plane, the graph of x ≥ 1 would be a solid vertical line with the region to the right of the line shaded. The graph of x > 1 would also have the area to the right of the vertical line shaded, but the line would be a dotted line to show that the boundary is not included.
Sketch the graphs for each of the systems of inequalities, showing only the common solution.
1. 
2. 
3. 
Absolute value equations
Most students learn the concept of absolute value as the magnitude of the number without the sign. For example, |5| = 5 and |−5| = 5. When these numbers (5 and −5) are graphed on a number line, you can see that both are 5 units from the origin. In fact, the geometric definition for absolute value is the distance a point is from the origin on the number line. This definition will prove helpful in solving absolute value equations and inequalities.
Solve each of the following absolute value equations.
1. |x + 2| = 3
2. |x − 5| = 6
3. |2x + 3| = 7
4. |5x − 8| = 7
5. |8 − 4x| = 12
6. |13 − 2x| = 7
Absolute value inequalities
If |x| = 5 represents those points that are exactly 5 units from the origin on the number line, then it makes sense that |x| > 5 represents those points that are more than 5 units from the origin, and |x| = 5 represents those points that are less than 5 units from the origin. That is, the solution to |x| > 5 is x > 5 or x < −5, while the solution to |x| = 5 is −5 = x = 5.
Solve each of the following absolute value inequalities.
1. |x − 4| ≤ 5
2. |x + 3| > 2
3. |4x − 3| ≥ 7
4. |7 − 3x| = 4
Write an absolute value inequality that describes each of the sets graphed below.
5. 
6. 