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Holt Algebra 1
3-Ext Solving Absolute-Value Inequalities3-Ext Solving Absolute-Value Inequalities
Holt Algebra 1
Lesson Presentation
Holt Algebra 1
3-Ext Solving Absolute-Value Inequalities
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 written as –5 < x < 5, which is the compound inequality x > –5 AND x < 5.
Holt Algebra 1
3-Ext Solving Absolute-Value Inequalities
Holt Algebra 1
3-Ext Solving Absolute-Value Inequalities
Example 1A: Solving Absolute-Value Inequalities Involving <
Solve the inequality and graph the solutions. Then write the solutions as a compound inequality.
|x| + 2 ≤ 6
|x| + 2 ≤ 6–2 –2
|x| ≤ 4
x ≥ –4 AND x ≤ 4
–5 –4 –3 –2 –1 0 1 2 3 4 5
4 units 4 units
–4 ≤ x ≤ 4
Since 2 is added to |x|, subtract 2 from both sides to undo the addition.
Think, “The distance from x to 0 is less than or equal to 4 units.”
Write as a compound inequality.
Holt Algebra 1
3-Ext Solving Absolute-Value Inequalities
|x| – 5 < –4
|x| – 5 < –4+5 +5|x| < 1
–1 < x AND x < 1
–1 < x < 1
Since 5 is subtracted from |x|, add 5 to both sides to undo the subtraction.
Think, “The distance from x to 0 is less than 1unit.”
x is between –1 and 1.Write as a compound inequality.
–2 –1 0 1 2
unit1
unit1
Example 1B: Solving Absolute-Value Inequalities Involving <
Solve the inequality and graph the solutions. Then write the solutions as a compound inequality.
Holt Algebra 1
3-Ext Solving Absolute-Value Inequalities
|x + 4| – 1.5 < 3.5
|x + 4| – 1.5 < 3.5+1.5 +1.5
|x + 4| < 5
Since 1.5 is subtracted from |x + 4|, add 1.5 to both sides to undo the subtraction.
Think, “The distance from x to –4 is less than 5 units.”
x + 4 > –5 AND x + 4 < 5 x + 4 is between –5 and 5.
–5 –4 –3 –2 –1 0 1 2 3 4 5
Example 1C: Solving Absolute-Value Inequalities Involving <
–4 –4 –4 –4
x > –9 AND x < 1
Solve the inequality and graph the solutions. Then write the solutions as a compound inequality.
5 units 5 units
Holt Algebra 1
3-Ext Solving Absolute-Value Inequalities
x > –9 AND x < 1
–9 < x < 1
–10 –8 –6 –4 –2 0 2 4 6 8 10
Write as a compound inequality.
Example 1C Continued
Holt Algebra 1
3-Ext Solving Absolute-Value Inequalities
Check It Out! Example 1a
|x| + 12 < 15
|x| + 12 < 15– 12 –12
|x| < 3
Since 12 is added to |x|, subtract 12 from both sides to undo the addition.
Think, “The distance from x to 0 is less than 3 units.”
x is between –3 and 3.x > –3 AND x < 3
–3 < x < 3
Write as a compound inequality.
Solve the inequality and graph the solutions. Then write the solutions as a compound inequality.
–5 –4 –3 –2 –1 0 1 2 3 4 5
3 units 3 units
Holt Algebra 1
3-Ext Solving Absolute-Value Inequalities
Check It Out! Example 1b
|x| – 6 < –5
|x| – 6 < –5+ 6 +6
|x| < 1
Since 6 is subtracted from |x|, add 6 to both sides to undo the subtraction.
Think, “The distance from x to 0 is less than 1 unit.”–2 –1 0 1 2
x > –1 AND x < 1
–1 < x < 1
x is between –1 and 1.
Write as a compound inequality.
Solve the inequality and graph the solutions. Then write the solutions as a compound inequality.
1 unit 1 unit
Holt Algebra 1
3-Ext Solving Absolute-Value Inequalities
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 written as the compound inequality x < –5 OR x > 5.
Holt Algebra 1
3-Ext Solving Absolute-Value Inequalities
Holt Algebra 1
3-Ext Solving Absolute-Value Inequalities
Solve the inequality and graph the solutions. Then write the solutions as a compound inequality.
Example 2A: Solving Absolute-Value Inequalities Involving >
|x| + 2 > 7|x| + 2 > 7
– 2 –2
|x| > 5
x < –5 OR x > 5
Since 2 is added to |x|, subtract 2 from both sides to undo the addition.
Write as a compound inequality.
–10 –8 –6 –4 –2 0 2 4 6 8 10
5 units 5 units
Holt Algebra 1
3-Ext Solving Absolute-Value Inequalities
Solve the inequality and graph the solutions. Then write the solutions as a compound inequality.
|x| – 12 ≥ –8|x| – 12 ≥ –8
+ 12 +12
|x| ≥ 4
–10 –8 –6 –4 –2 0 2 4 6 8 10
x ≤ –4 OR x ≥ 4
Since 12 is subtracted from |x|, add 12 to both sides to undo the subtraction.
Write as a compound inequality.
Example 2B: Solving Absolute-Value Inequalities Involving >
4 units 4 units
Holt Algebra 1
3-Ext Solving Absolute-Value Inequalities
Solve the inequality and graph the solutions. Then write the solutions as a compound inequality.
Example 2C: Solving Absolute-Value Inequalities Involving >
|x + 3| – 5 > 9
|x + 3| – 5 > 9+ 5 +5
|x + 3| > 14
Since 5 is subtracted from |x + 3|, add 5 to both sides to undo the subtraction.
–16 –12 –8 –4 0 4 8 12 16
x + 3 < –14 OR x + 3 > 14
14 units 14 units
Holt Algebra 1
3-Ext Solving Absolute-Value Inequalities
– 3 –3 –3 –3
x < –17 OR x > 11
Solve the two inequalities.
–24 –20 –16 –12 –8 –4 0 4 8 12 16
–17 11Graph.
Example 2C Continued
x + 3 < –14 OR x + 3 > 14
Holt Algebra 1
3-Ext Solving Absolute-Value Inequalities
Check It Out! Example 2a
|x| + 10 ≥ 12
|x| + 10 ≥ 12– 10 –10
|x| ≥ 2
–5 –4 –3 –2 –1 0 1 2 3 4 5
x ≤ –2 OR x ≥ 2 Write as a compound inequality.
Since 10 is added to |x|, subtract 10 from both sides to undo the addition.
Solve the inequality and graph the solutions. Then write the solutions as a compound inequality.
2 units 2 units
Holt Algebra 1
3-Ext Solving Absolute-Value Inequalities
Check It Out! Example 2b
|x| – 7 > –1
|x| – 7 > –1+7 +7
|x| > 6
–10 –8 –6 –4 –2 0 2 4 6 8 10
x < –6 OR x > 6
Since 7 is subtracted from |x|, add 7 to both sides to undo the subtraction.
Write as a compound inequality.
Solve the inequality and graph the solutions. Then write the solutions as a compound inequality.
6 units 6 units
Holt Algebra 1
3-Ext Solving Absolute-Value Inequalities
Check It Out! Example 2c
–5 –4 –3 –2 –1 0 1 2 3 4 5
Since is added to |x + 2 |, subtract from both sides to undo the addition.
OR
Solve the inequality and graph the solutions. Then write the solutions as a compound inequality.
3.5 units 3.5 units
Holt Algebra 1
3-Ext Solving Absolute-Value Inequalities
Check It Out! Example 2c Continued
OR
OR
–10 –8 –6 –4 –2 0 2 4 6 8 10
1
Solve the two inequalities.
Graph.