Holt McDougal Algebra 1 2-7 Solving Absolute-Value Inequalities

Holt McDougal Algebra 1

2-7 Solving Absolute-Value Inequalities





Solve compound inequalities in one variable involving absolute-value expressions.

Objectives



When an inequality contains an absolute-value expression, it can be written as a compound inequality. The inequality |x| < 5 describes all real numbers whose distance from 0 is less than 5 units. The solutions are all numbers between –5 and 5, so |x|< 5 can be rewritten as –5 < x < 5, or as x > –5 AND x < 5.



Additional Example 1A: Solving Absolute-Value Inequalities Involving <

Solve the inequality and graph the solutions.

|x|– 3 < –1

x > –2 AND x < 2

-2<x<2

–2 –1 0 1 2

2 units 2 units



|x – 1| ≤ 2

Additional Example 1B: Solving Absolute-Value Inequalities Involving <


x ≥ –1 x ≤ 3 AND

–2 –1 0 1 2 3–3

-1≤x≤3



Just as you do when solving absolute-value equations, you first isolate the absolute-value expression when solving absolute-value inequalities.

Helpful Hint



Check It Out! Example 1a


2|x| ≤ 6

x ≥ –3 AND x ≤ 3

-3≤x≤3

–2 –1 0 1 2

3 units 3 units

–3 3


2-7 Solving Absolute-Value InequalitiesCheck It Out! Example 1b

|x + 3|– 4.5 ≤ 7.5

Solve each inequality and graph the solutions.

–20 –15 –10 –5 0 5 10 15



The inequality |x| > 5 describes all real numbers whose distance from 0 is greater than 5 units. The solutions are all numbers less than –5 or greater than 5. The inequality |x| > 5 can be rewritten as the compound inequality x < –5 OR x > 5.




Additional Example 2A: Solving Absolute-Value Inequalities Involving >

|x| + 14 ≥ 19

|x| ≥ 5

x ≤ –5 OR x ≥ 5

–10 –8 –6 –4 –2 0 2 4 6 8 10

5 units 5 units




3 + |x + 2| > 5

Additional Example 2B: Solving Absolute-Value Inequalities Involving >

x < –4 OR x > 0

–10 –8 –6 –4 –2 0 2 4 6 8 10


2-7 Solving Absolute-Value InequalitiesCheck It Out! Example 2a

|x| + 10 ≥ 12

–5 –4 –3 –2 –1 0 1 2 3 4 5

x ≤ –2 OR x ≥ 2


2 units 2 units



Check It Out! Example 2bSolve the inequality and graph the solutions.

x ≤ –6 or

x ≥ 1

–7 –6 –5 –4 –3 0 1 2 3–2 –1



Additional Example 3: ApplicationA pediatrician recommends that a baby’s bath water be 95°F, but it is acceptable for the temperature to vary from this amount by as much as 3°F. Write and solve an absolute-value inequality to find the range of acceptable temperatures. Graph the solutions.

The range of acceptable temperature is 92 ≤ t ≤ 98.

98 10096949290



Check It Out! Example 3

A dry-chemical fire extinguisher should be pressurized to 125 psi, but it is acceptable for the pressure to differ from this value by at most 75 psi. Write and solve an absolute-value inequality to find the range of acceptable pressures. Graph the solution.

The range of pressure is 50 ≤ p ≤ 200.

200 225175150125100755025



When solving an absolute-value inequality, you may get a statement that is true for all values of the variable. In this case, all real numbers are solutions of the original inequality. If you get a false statement when solving an absolute-value inequality, the original inequality has no solutions.



Additional Example 4A: Special Cases of Absolute-Value Inequalities

Solve the inequality.

|x + 4|– 5 > – 8

|x + 4| > –3

All real numbers are solutions.


2-7 Solving Absolute-Value InequalitiesAdditional Example 4B: Special Cases of Absolute-Value

Inequalities


|x – 2| + 9 < 7

The inequality has no solutions.



An absolute value represents a distance, and distance cannot be less than 0.

Remember!



Check It Out! Example 4a


|x| – 9 ≥ –11

All real numbers are solutions.



Check It Out! Example 4b


4|x – 3.5| ≤ –8

The inequality has no solutions.


2-7 Solving Absolute-Value InequalitiesLesson Quiz: Part I


1. 3|x| > 15 x < –5 or x > 5

0–5–10 5 10

2. |x + 3| + 1 < 3 –5 < x < –1

–2 0–1–3–4–5–6

3. A number, n, is no more than 7 units away from 5. Write and solve an inequality to show the range of possible values for n.|n– 5| ≤ 7; –2 ≤ n ≤ 12



Solve each inequality.

4. |3x| + 1 < 1

5. |x + 2| – 3 ≥ – 6 all real numbers

no solutions

Lesson Quiz: Part II

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Holt McDougal Algebra 1 2-7 Solving Absolute-Value Inequalities