Absolute value inequalities worksheet with problems and number lines for solving.

You're working on Absolute Value Inequalities: Extra Terms — this means each inequality has an absolute value expression *plus* some extra constant terms added or subtracted, and you need to solve for `x`.

The key strategy is:

> Isolate the absolute value expression first, then apply the rules for solving absolute value inequalities.

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📌 General Rules for Absolute Value Inequalities:

For any expression `|A|` and constant `c > 0`:

- If `|A| < c`, then `-c < A < c`
- If `|A| ≤ c`, then `-c ≤ A ≤ c`
- If `|A| > c`, then `A < -c` OR `A > c`
- If `|A| ≥ c`, then `A ≤ -c` OR `A ≥ c`

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Let’s solve each problem step-by-step.

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## ✔ Problem 1: `2|x + 4| + 9 > 11`

Step 1: Subtract 9 from both sides
→ `2|x + 4| > 2`

Step 2: Divide by 2
→ `|x + 4| > 1`

Step 3: Apply “greater than” rule
→ `x + 4 < -1` OR `x + 4 > 1`

Step 4: Solve each
→ `x < -5` OR `x > -3`

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

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## ✔ Problem 2: `3|x + 8| + 4 ≤ 6`

Step 1: Subtract 4
→ `3|x + 8| ≤ 2`

Step 2: Divide by 3
→ `|x + 8| ≤ 2/3`

Step 3: Apply “less than or equal” rule
→ `-2/3 ≤ x + 8 ≤ 2/3`

Step 4: Subtract 8
→ `-8 - 2/3 ≤ x ≤ -8 + 2/3`
→ `-26/3 ≤ x ≤ -22/3`

✔ Solution: `-26/3 ≤ x ≤ -22/3` (or approximately `-8.67 ≤ x ≤ -7.33`)
Graph: Closed circles at -26/3 and -22/3, line segment between them.

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## ✔ Problem 3: `-4|x - 1| - 7 < -1`

Step 1: Add 7 to both sides
→ `-4|x - 1| < 6`

Step 2: Divide by -4 → flip inequality sign!
→ `|x - 1| > -6/4` → `|x - 1| > -3/2`

BUT — absolute value is always ≥ 0, so `|x - 1| > -3/2` is always true for all real x.

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

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## ✔ Problem 4: `|2x - 1| - 8 ≥ 11`

Step 1: Add 8
→ `|2x - 1| ≥ 19`

Step 2: Apply “≥” rule
→ `2x - 1 ≤ -19` OR `2x - 1 ≥ 19`

Step 3: Solve each
→ `2x ≤ -18` → `x ≤ -9`
→ `2x ≥ 20` → `x ≥ 10`

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

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## ✔ Problem 5: `|2x + 2| - 2 ≤ 22`

Step 1: Add 2
→ `|2x + 2| ≤ 24`

Step 2: Apply “≤” rule
→ `-24 ≤ 2x + 2 ≤ 24`

Step 3: Subtract 2
→ `-26 ≤ 2x ≤ 22`

Step 4: Divide by 2
→ `-13 ≤ x ≤ 11`

✔ Solution: `-13 ≤ x ≤ 11`
Graph: Closed circles at -13 and 11, line segment between.

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## ✔ Problem 6: `6|5 + x| - 35 > 16`

Step 1: Add 35
→ `6|5 + x| > 51`

Step 2: Divide by 6
→ `|x + 5| > 51/6 = 17/2 = 8.5`

Step 3: Apply “>” rule
→ `x + 5 < -8.5` OR `x + 5 > 8.5`

Step 4: Subtract 5
→ `x < -13.5` OR `x > 3.5`

✔ Solution: `x < -13.5` or `x > 3.5`
Graph: Open circles at -13.5 and 3.5, arrows left and right.

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## ✔ Problem 7: `12|2x - 1| - 15 < 21`

Step 1: Add 15
→ `12|2x - 1| < 36`

Step 2: Divide by 12
→ `|2x - 1| < 3`

Step 3: Apply “<” rule
→ `-3 < 2x - 1 < 3`

Step 4: Add 1
→ `-2 < 2x < 4`

Step 5: Divide by 2
→ `-1 < x < 2`

✔ Solution: `-1 < x < 2`
Graph: Open circles at -1 and 2, line segment between.

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## ✔ Problem 8: `-|5x - 5| + 22 ≥ 2`

Step 1: Subtract 22
→ `-|5x - 5| ≥ -20`

Step 2: Multiply both sides by -1 → flip inequality
→ `|5x - 5| ≤ 20`

Step 3: Apply “≤” rule
→ `-20 ≤ 5x - 5 ≤ 20`

Step 4: Add 5
→ `-15 ≤ 5x ≤ 25`

Step 5: Divide by 5
→ `-3 ≤ x ≤ 5`

✔ Solution: `-3 ≤ x ≤ 5`
Graph: Closed circles at -3 and 5, line segment between.

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## ✔ Problem 9: `4|9 - 3x| - 2 ≤ 10`

Step 1: Add 2
→ `4|9 - 3x| ≤ 12`

Step 2: Divide by 4
→ `|9 - 3x| ≤ 3`

Note: `|9 - 3x| = |-(3x - 9)| = |3x - 9|`, so same as `|3x - 9| ≤ 3`

Step 3: Apply “≤” rule
→ `-3 ≤ 3x - 9 ≤ 3`

Step 4: Add 9
→ `6 ≤ 3x ≤ 12`

Step 5: Divide by 3
→ `2 ≤ x ≤ 4`

✔ Solution: `2 ≤ x ≤ 4`
Graph: Closed circles at 2 and 4, line segment between.

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## ✔ Problem 10: `|-8 - 4x| - 15 > 17`

Step 1: Add 15
→ `|-8 - 4x| > 32`

Note: `|-8 - 4x| = |-(8 + 4x)| = |8 + 4x| = |4x + 8|`

So: `|4x + 8| > 32`

Step 2: Apply “>” rule
→ `4x + 8 < -32` OR `4x + 8 > 32`

Step 3: Solve each
→ `4x < -40` → `x < -10`
→ `4x > 24` → `x > 6`

✔ Solution: `x < -10` or `x > 6`
Graph: Open circles at -10 and 6, arrows left and right.

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## 🧾 Final Answer Summary:

1. `x < -5` or `x > -3`
2. `-26/3 ≤ x ≤ -22/3`
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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📌 Tip for graphing: Use open circles for strict inequalities (`<`, `>`), closed circles for inclusive ones (`≤`, `≥`). Shade accordingly.

Let me know if you want the graphs drawn or need help with a specific one!