1. Solving Absolute Value Equations and Inequalities
Section 8.5
Solving Absolute Value Equations and Inequalities

2. Objectives Solve equations of the form | X | = k
Solve equations with two absolute values
Solve inequalities of the form | X | < k
Solve inequalities of the form | X | > k

3. Objective 1: Solve Equations of the Form | X | = k
To solve the absolute value equation | x | = 5, we must find all real numbers x whose distance from 0 on the number line is 5. There are two such numbers: 5 and –5. It follows that the solutions of | x | = 5 are 5 and –5 and the solution set is {5, –5}.
Solving Absolute Value Equations For any positive number k and any algebraic expression X: To solve | X | = k, solve the equivalent compound equation X = k or X = –k. This is called a compound equation because it consists of two equations joined with the word or.

4. EXAMPLE 1
Solve: a. | s | = b. | 3x – 2 | = 5
c. | 10 – x | = –40
Strategy To solve each of these absolute value equations, we will write and solve an equivalent compound equation. We will solve the third equation by inspection.
Why All three of the equations are of the form
| X | = k. However, the standard method for solving absolute value equations cannot be applied to
| 10 – x | = –40, because k is negative.

5. EXAMPLE 1
Solve: a. | s | = b. | 3x – 2 | = 5
c. | 10 – x | = –40
Solution
a. The absolute value equation | s | = is equivalent to the compound equation
s = or s = –0.003
Therefore, the solutions of | s | = are and
–0.003 and the solution set is {0.003, –0.003}.
b. The absolute value equation | 3x – 2 | = 5 is equivalent to the compound equation 3x – 2 = 5 or 3x – 2 = –5.

6. EXAMPLE 1 Solution Solve: a. | s | = 0.003 b. | 3x – 2 | = 5
c. | 10 – x | = –40
Solution
Now we solve each equation for x:
The result must be checked separately to see whether each of them produces a true statement. We substitute for x and then –1 in the original equation.
The resulting true statement indicate that the equation has
two solutions: and –1. The
solution set is

7. EXAMPLE 1
Solve: a. | s | = b. | 3x – 2 | = 5
c. | 10 – x | = –40
Solution
c. Since an absolute value can never be negative, there are no real numbers x that make | 10 – x | = –40 true. The equation has no solution and the solution set is .

8. Objective 2: Solve Equations with Two Absolute Values
Equations can contain two absolute value expressions. To develop a strategy to solve them, consider the following examples.
Solving Equations with Two Absolute Values For any algebraic expressions X and Y:
To solve | X | = | Y |, solve the compound equation X = Y or
X = –Y.

9. EXAMPLE 5
Solve: | 5x + 3 | = | 3x + 25 |
Strategy To solve this equation, we will write and then solve an equivalent compound equation.
Why We can use this approach because the equation is of the form | X | = | Y |.

10. EXAMPLE 5
Solve: | 5x + 3 | = | 3x + 25 |
Solution
The equation | 5x + 3 | = | 3x + 25 |, with the two absolute value expressions, is equivalent to the following compound equation:
Solve each equation.
Simplify the fraction.
Verify that both solutions, 11 and , check by substituting them into the original equation.

11. Objective 3: Solve Inequalities of the Form | X | < k
To solve the absolute value inequality | x | < 5, we must find all real numbers x whose distance from 0 on the number line is less than 5. From the graph, we see that there are many such numbers.
Solving | X | < k and | X | ≤ k For any positive number k and any algebraic expression X:
To solve | X | < k, solve the equivalent double inequality –k < X < k.
To solve | X | ≤ k, solve the equivalent double inequality –k ≤ X ≤ k.

12. EXAMPLE 6
Solve | 2x – 3 | < 9 and graph the solution set.
Strategy To solve this absolute value inequality, we will write and solve an equivalent double inequality.
Why We can use this approach because the inequality is of the form | X | < k, and k is positive.

13. EXAMPLE 6
Solve | 2x – 3 | < 9 and graph the solution set.
Solution
The absolute value inequality | 2x – 3 | < 9 is equivalent to the double inequality –9 < 2x – 3 < 9, which we can solve for x:
Any number between –3 and 6 is in the solution set, which can be
written as {x| –3 < x < 6}. This is the interval (–3, 6); its graph is shown
below.

14. Objective 4: Solve Inequalities of the Form | X | > k
Solving | X | > k and | X | ≥ k For any positive number k and any algebraic expression X:
To solve | X | > k, solve the equivalent compound inequality X < –k or X > k.
To solve | X | ≥ k, solve the equivalent compound inequality X ≤ –k or X ≥ k.

15. EXAMPLE 9
Solve and graph the solution set.
Strategy To solve this absolute value inequality, we will write and solve an equivalent compound inequality.
Why We can use this approach because the inequality is of the form | X | ≥ k, and k is positive.

16. EXAMPLE 9
Solve and graph the solution set.
Solution
The absolute value inequality is equivalent to the compound inequality
Now we solve each inequality for x:
The solution set is the union of two intervals:
(–∞, –27] U [33, ∞).
Its graph appears on the right.