Absolute value inequalities worksheet with problems and number lines for graphing solutions.

You’ve uploaded a worksheet titled “Absolute Value Inequalities: Extra Terms”, which asks you to solve 10 inequalities involving absolute values and extra constant terms. I’ll solve each one step-by-step and explain the process.

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🔑 Key Concept:

To solve absolute value inequalities, isolate the absolute value expression first, then apply the rules:

- If |A| > B (B > 0), then A < -B or A > B
- If |A| < B (B > 0), then -B < A < B
- If |A| ≥ B → A ≤ -B or A ≥ B
- If |A| ≤ B → -B ≤ A ≤ B

Also remember: if you multiply or divide by a negative number, flip the inequality sign.

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## ✔ Problem 1:
2|x + 4| + 9 > 11

Subtract 9:
→ 2|x + 4| > 2
Divide by 2:
→ |x + 4| > 1

Apply rule for “>”:
→ x + 4 < -1 OR x + 4 > 1
→ x < -5 OR x > -3

✔ Solution: x < -5 or x > -3
Number line: Open circles at -5 and -3, arrows going left from -5 and right from -3.

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## ✔ Problem 2:
2|x + 8| + 4 ≤ 6

Subtract 4:
→ 2|x + 8| ≤ 2
Divide by 2:
→ |x + 8| ≤ 1

Apply rule for “≤”:
→ -1 ≤ x + 8 ≤ 1
Subtract 8:
→ -9 ≤ x ≤ -7

✔ Solution: -9 ≤ x ≤ -7
Number line: Closed circles at -9 and -7, shaded between them.

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## ✔ Problem 3:
-4|x - 1| - 7 < -1

Add 7 to both sides:
→ -4|x - 1| < 6
Divide by -4 → flip inequality:
→ |x - 1| > -3/2

Wait — absolute value is always ≥ 0, so |x - 1| > -1.5 is ALWAYS TRUE because absolute value can’t be negative.

✔ Solution: All real numbers → (-∞, ∞)
Number line: Entire line shaded.

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## ✔ Problem 4:
|2x - 1| - 8 ≥ 11

Add 8:
→ |2x - 1| ≥ 19

Apply “≥” rule:
→ 2x - 1 ≤ -19 OR 2x - 1 ≥ 19
Solve each:
→ 2x ≤ -18 → x ≤ -9
→ 2x ≥ 20 → x ≥ 10

✔ Solution: x ≤ -9 or x ≥ 10
Number line: Closed circles at -9 and 10, arrows left from -9 and right from 10.

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## ✔ Problem 5:
|2x + 2| - 2 ≤ 22

Add 2:
→ |2x + 2| ≤ 24

Apply “≤” rule:
→ -24 ≤ 2x + 2 ≤ 24
Subtract 2:
→ -26 ≤ 2x ≤ 22
Divide by 2:
→ -13 ≤ x ≤ 11

✔ Solution: -13 ≤ x ≤ 11
Number line: Closed circles at -13 and 11, shaded between.

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## ✔ Problem 6:
6|5 + x| - 35 > 16

Add 35:
→ 6|5 + x| > 51
Divide by 6:
→ |x + 5| > 51/6 = 8.5

Apply “>” rule:
→ x + 5 < -8.5 OR x + 5 > 8.5
→ x < -13.5 OR x > 3.5

✔ Solution: x < -13.5 or x > 3.5
Number line: Open circles at -13.5 and 3.5, arrows left/right.

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## ✔ Problem 7:
12|2x - 1| - 15 < 21

Add 15:
→ 12|2x - 1| < 36
Divide by 12:
→ |2x - 1| < 3

Apply “<” rule:
→ -3 < 2x - 1 < 3
Add 1:
→ -2 < 2x < 4
Divide by 2:
→ -1 < x < 2

✔ Solution: -1 < x < 2
Number line: Open circles at -1 and 2, shaded between.

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## ✔ Problem 8:
-|5x - 5| + 22 ≥ 2

Subtract 22:
→ -|5x - 5| ≥ -20
Multiply both sides by -1 → flip inequality:
→ |5x - 5| ≤ 20

Apply “≤” rule:
→ -20 ≤ 5x - 5 ≤ 20
Add 5:
→ -15 ≤ 5x ≤ 25
Divide by 5:
→ -3 ≤ x ≤ 5

✔ Solution: -3 ≤ x ≤ 5
Number line: Closed circles at -3 and 5, shaded between.

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## ✔ Problem 9:
4|9 - 3x| - 2 ≤ 10

Add 2:
→ 4|9 - 3x| ≤ 12
Divide by 4:
→ |9 - 3x| ≤ 3

Note: |9 - 3x| = | -3x + 9 | = |3x - 9| (absolute value is symmetric)
So we can write: |3x - 9| ≤ 3

Apply “≤” rule:
→ -3 ≤ 3x - 9 ≤ 3
Add 9:
→ 6 ≤ 3x ≤ 12
Divide by 3:
→ 2 ≤ x ≤ 4

✔ Solution: 2 ≤ x ≤ 4
Number line: Closed circles at 2 and 4, shaded between.

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## ✔ Problem 10:
|-8 - 4x| - 15 > 17

Add 15:
→ |-8 - 4x| > 32
Note: |-8 - 4x| = |-(4x + 8)| = |4x + 8|
So: |4x + 8| > 32

Apply “>” rule:
→ 4x + 8 < -32 OR 4x + 8 > 32
Solve each:
→ 4x < -40 → x < -10
→ 4x > 24 → x > 6

✔ Solution: x < -10 or x > 6
Number line: Open circles at -10 and 6, arrows left/right.

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## 📝 Final Answer Summary:

1. x < -5 or x > -3
2. -9 ≤ x ≤ -7
3. All real numbers
4. x ≤ -9 or x ≥ 10
5. -13 ≤ x ≤ 11
6. x < -13.5 or x > 3.5
7. -1 < x < 2
8. -3 ≤ x ≤ 5
9. 2 ≤ x ≤ 4
10. x < -10 or x > 6

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Let me know if you’d like these graphed on number lines or need help with word problems or more complex cases!