# Absolute Value & Inequalities

## Definitions

An inequality compares two values.

• Inequality - A comparison of two values or expressions.
For example, 15x < 45 is an inequality whereas x = 3 is an equation.
• Equation - A statement declaring the equality of two expressions.
For example, 25x = 50 is an equation whereas 25x > 50 is an inequality.

## Operating With Inequalities: Absolute Value

Just as traditional equations with absolute value term within them usually have two solutions, so inequalities with an absolute value term within them typically have two solutions. In order to solve an inequality with an absolute value term, isolate the absolute value, set the expression within the absolute value brackets equal to a positive and negative value, and solve for the unknown variable.

1.) Isolate the absolute value term
10 + |2x + 15| < 55
10 - 10 + |2x + 15| < 55 - 10
|2x + 15| < 45

2.) Set the expression within the absolute value term equal to both a positive and negative value

Positive: 2x + 15 < 45
2x + 15 -15 < 45 - 15
2x < 30
x < 15

Negative: -(2x + 15) < 45
-2x - 15 < 45
-2x - 15 + 15 < 45 + 15
-2x < 60
x > -30 [The sign of the inequality is flipped because we divide by a negative number]

As a check, if x = 30 (outside the solution set), 10 + |2x + 15| = 85, which does not fit the inequality (i.e., 85 is not less than 55). However, if x = -10 (inside the solution set), 10 + |2x + 15| = 15, which does fit the inequality (i.e., 15 is less than 55).

## Multiple Inequalities

The process for solving multiple inequalities with absolute value is the same as the process for solving multiple inequalities without absolute value. In solving multiple simultaneous inequalities with absolute value, the most important part is to solve each inequality separately and then combine them.

If |x + 7| < 14 and |4x - 4| > 16, what is the range of possible values for x?

1.) Solve each inequality alone.
|x + 7| < 14
Positive: x + 7 < 14
x + 7 - 7 < 14 - 7
x < 7
Negative: -1(x + 7) < 14
-x - 7 < 14
-x - 7 + 7 < 14 + 7
-x < 21
x > -21

|4x - 4| > 16
Positive: 4x - 4 > 16
4x - 4 + 4 > 16 + 4
4x > 20
x > 5
Negative: -1(4x - 4) > 16
-4x + 4 > 16
-4x + 4 - 4 > 16 - 4
-4x > 12
x < -3

2.) Combine each inequality and find the overlap (i.e., the areas where each inequality is satisfied--this area is the solution).
x < 7
x > -21
x > 5
x < -3

There are two areas where each inequality is satisfied: (1) -21 < x < - 3 (2) 5 < x < 7

For many students, the above set of inequalities can best be understood graphically. The solution to the set of inequalities is the overlapping graphical area. The red lines represent the solutions to the |4x - 4| > 16 inequality while the blue lines represent the solution to the |x + 7| < 14 inequality. 