Holt McDougal Algebra 1 2-7 Solving Absolute-Value Inequalities Solve compound inequalities in one...
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Transcript of Holt McDougal Algebra 1 2-7 Solving Absolute-Value Inequalities Solve compound inequalities in one...
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Solve compound inequalities in one variable involving absolute-value expressions.
Objectives
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Find all numbers whose absolute value is less than 5.
Absolute value inequality:
Compound inequality:
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Additional Example 1A: Solving Absolute-Value Inequalities Involving <
Solve the inequality and graph the solutions.
|x|– 3 < –1
x > –2 AND x < 2
Since 3 is subtracted from |x|, add 3 to both sides to undo the subtraction.
+3 +3|x| < 2
|x|– 3 < –1
Write as a compound inequality.
–2 –1 0 1 2
2 units 2 units
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
|x – 1| ≤ 2
Write as a compound inequality.
Additional Example 1B: Solving Absolute-Value Inequalities Involving <
Solve the inequality and graph the solutions.
x – 1 ≥ –2 AND x – 1 ≤ 2 Solve each inequality.+1 +1 +1 +1
x ≥ –1 x ≤ 3AND Write as a compound inequality.
–2 –1 0 1 2 3–3
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Check It Out! Example 1a
Solve the inequality and graph the solutions.
2|x| ≤ 6
x ≥ –3 AND x ≤ 3
Since x is multiplied by 2, divide both sides by 2 to undo the multiplication.|x| ≤ 3
Write as a compound inequality.
2|x| ≤ 62 2
–2 –1 0 1 2
3 units 3 units
–3 3
Holt McDougal Algebra 1
2-7 Solving Absolute-Value InequalitiesCheck It Out! Example 1b
|x + 3|– 4.5 ≤ 7.5Since 4.5 is subtracted from |
x + 3|, add 4.5 to both sides to undo the subtraction.
Solve each inequality and graph the solutions.
+ 4.5 +4.5 |x + 3| ≤ 12
|x + 3|– 4.5 ≤ 7.5
x + 3 ≥ –12 AND x + 3 ≤ 12 –3 –3 –3 –3x ≥ –15 AND x ≤ 9
Write as a compound inequality.
–20 –15 –10 –5 0 5 10 15
Subtract 3 from both sides of each inequality.
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Find all numbers whose absolute value is greater than 5.
Absolute value inequality:
Compound inequality:
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Solve the inequality and graph the solutions.
Additional Example 2A: Solving Absolute-Value Inequalities Involving >
|x| + 14 ≥ 19
|x| ≥ 5
x ≤ –5 OR x ≥ 5
Since 14 is added to |x|, subtract 14 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
– 14 –14|x| + 14 ≥ 19
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Solve the inequality and graph the solutions.
3 + |x + 2| > 5Since 3 is added to |x + 2|,
subtract 3 from both sides to undo the addition.
Write as a compound inequality. Solve each inequality.
Additional Example 2B: Solving Absolute-Value Inequalities Involving >
|x + 2| > 2
– 3 – 33 + |x + 2| > 5
x + 2 < –2 OR x + 2 > 2–2 –2 –2 –2
x < –4 OR x > 0 Write as a compound inequality.
–10 –8 –6 –4 –2 0 2 4 6 8 10
Holt McDougal Algebra 1
2-7 Solving Absolute-Value InequalitiesCheck 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 each inequality and graph the solutions.
2 units 2 units
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Check It Out! Example 2b
Since is added to |x + 2 |, subtract from both sides to undo the addition.
OR
Solve the inequality and graph the solutions.
x ≤ –6 x ≥ 1
Write as a compound inequality.
Write as a compound inequality. Solve each inequality.
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Check It Out! Example 2b Continued
Solve the inequality and graph the solutions.
–7 –6 –5 –4 –3 0 1 2 3–2 –1
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Homework:Sec. 2-7 Practice B Wksht (1-8) & Sec. 2-7 Practice A Wksht (1-8)
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
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.
Let t represent the actual water temperature.
The difference between t and the ideal temperature is at most 3°F.
t – 95 ≤ 3
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Additional Example 3 Continued
t – 95 ≤ 3
|t – 95| ≤ 3
t – 95 ≥ –3 AND t – 95 ≤ 3 Solve the two inequalities.+95 +95 +95 +95
t ≥ 92 AND t ≤ 98
The range of acceptable temperature is 92 ≤ t ≤ 98.
98 10096949290
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
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.
Let p represent the desired pressure.
The difference between p and the ideal pressure is at most 75 psi.
p – 125 ≤ 75
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Check It Out! Example 3 Continued
p – 125 ≤ 75
|p – 125| ≤ 75
p – 125 ≥ –75 AND p – 125 ≤ 75 Solve the two inequalities.+125 +125 +125 +125
p ≥ 50 AND p ≤ 200
The range of pressure is 50 ≤ p ≤ 200.
200 225175150125100755025
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
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.
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Additional Example 4A: Special Cases of Absolute-Value Inequalities
Solve the inequality.
|x + 4|– 5 > – 8
|x + 4|– 5 > – 8 + 5 + 5
|x + 4| > –3
Add 5 to both sides.
Absolute-value expressions are always nonnegative. Therefore, the statement is true for all real numbers.
All real numbers are solutions.
Holt McDougal Algebra 1
2-7 Solving Absolute-Value InequalitiesAdditional Example 4B: Special Cases of Absolute-Value
Inequalities
Solve the inequality.
|x – 2| + 9 < 7
|x – 2| + 9 < 7 – 9 – 9
|x – 2| < –2
Subtract 9 from both sides.
Absolute-value expressions are always nonnegative. Therefore, the statement is false for all values of x.
The inequality has no solutions.
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
An absolute value represents a distance, and distance cannot be less than 0.
Remember!
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Check It Out! Example 4a
Solve the inequality.
|x| – 9 ≥ –11
|x| – 9 ≥ –11 +9 ≥ +9
|x| ≥ –2Add 9 to both sides.
Absolute-value expressions are always nonnegative. Therefore, the statement is true for all real numbers.
All real numbers are solutions.
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Check It Out! Example 4b
Solve the inequality.
4|x – 3.5| ≤ –8
4|x – 3.5| ≤ –8
4 4|x – 3.5| ≤ –2 Absolute-value expressions are
always nonnegative. Therefore, the statement is false for all values of x.
Divide both sides by 4.
The inequality has no solutions.
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Homework:Sec. 2-7 Practice C & Problem
Solving Worksheets