| Absolute Value Linear Inequalities by Example |
-- Section 1.8 -- |
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Case 1 |
| x – a | < b means - b < x – a < b |
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Solve for x: | x – 2 | < 5 |
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Solve |
Step |
Graph |
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| x – 2 | < 5 |
Use this technique when |
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-5 < x – 2 < 5 |
|x| < a Û -a < x < a |
Click Link for Inequality Grapher |
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-5 + 2 < x – 2 + 2 < 5 + 2 |
Add 2 to each part | |
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-3 < x < 7 |
Inequality Notation | |
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[-3, 7] |
Interval Notation | |
| This means x has to be between -3 and 7 (x could be -3 or 7 because of the < symbol) | ||
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Solve for x: | x – 2 | < 3 |
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[Solution] |
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Solve for x: | 3x + 2 | < 7 |
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Solve |
Step |
Graph |
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| 3x + 2 | < 7 -7 < 3x + 2 < 7 -7 – 2 < 3x + 2 – 2 < 7 – 2 -9 < 3x < 5 -3 < x < 5/3 (-3, 5/3) |
Use this technique when |x| < a Û -a < x < a Subtract 2 from each part Divide each part by 3 Inequality Notation Interval Notation |
Click Link for Inequality Grapher. |
| This means x has to be between -3 and 5/3 (x could not be -3 or 5/3 because of the < symbol) | ||
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Solve for x:
| 4x +
1 | < 9 |
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[Solution] |
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Case 2 |
| x – a | > b means x – a < -b or x – a > b |
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Solve for x: | x + 2 | > 4 |
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Solve |
Step |
Graph |
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| x + 2 | > 4 |
Use this technique when | |
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x + 2 < -4 or x + 2 > 4 |
|x| > a Û x < -a or x > a |
Click Link for Inequality Grapher. |
| Solve each inequality separately. | ||
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x + 2 – 2 < -4 – 2 |
First inequality | |
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x < -6 |
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x + 2 – 2 > 4 – 2 |
Second Inequality | |
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x > 2 |
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x < -6 or x > 2 |
Inequality Notation | |
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Interval Notation | |
| This means x has to be less than or equal to -6 or greater than or equal to 2 | ||
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Solve for x: | x – 1 | > 5 |
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[Solution] |
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Solve for x: | 2x + 3 | > 3 |
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Solve |
Step |
Graph |
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| 2x + 3 | > 3 |
Use this
technique when |x| > a |
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2x + 3 < -3 or 2x + 3 > 3 |
Solve each inequality separately. | |
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Click Link for Inequality Grapher. |
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2x + 3 – 3 < -3 – 3 |
First inequality | |
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2x < -6 |
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x < -3 |
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2x + 3 – 3 > 3 – 3 |
Second Inequality | |
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2x > 0 |
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x > 0 |
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x < -3 or x > 0 |
Inequality Notation | |
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Interval Notation | |
| This means x has to be less than -3 or greater than 0 | ||
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Solve for x: | 4x – 1 | > 7 |
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[Solution] |
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Tutorials and Applets by
Joe McDonald
Community College of Southern Nevada
