Graphing Inequalities Worksheets with Answer Key - Free Printable

Let's solve each inequality step by step and then describe how to graph the solution on a number line. I'll go through all 10 problems.

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① $ 4(2x - 3) \geq -8 $

Step 1: Distribute
$ 8x - 12 \geq -8 $

Step 2: Add 12 to both sides
$ 8x \geq 4 $

Step 3: Divide by 8
$ x \geq \frac{1}{2} $

✔ Solution: $ x \geq \frac{1}{2} $
➡️ Graph: Closed circle at $ \frac{1}{2} $, shaded to the right.

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② $ 4x + 6 \leq 18 $

Step 1: Subtract 6
$ 4x \leq 12 $

Step 2: Divide by 4
$ x \leq 3 $

✔ Solution: $ x \leq 3 $
➡️ Graph: Closed circle at 3, shaded to the left.

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③ $ -10 < 6x + 2 $

Step 1: Subtract 2
$ -12 < 6x $

Step 2: Divide by 6
$ -2 < x $ → or $ x > -2 $

✔ Solution: $ x > -2 $
➡️ Graph: Open circle at -2, shaded to the right.

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④ $ 4(x - 2) > 20 $

Step 1: Distribute
$ 4x - 8 > 20 $

Step 2: Add 8
$ 4x > 28 $

Step 3: Divide by 4
$ x > 7 $

✔ Solution: $ x > 7 $
➡️ Graph: Open circle at 7, shaded to the right.

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⑤ $ 6x + 7 < 37 $

Step 1: Subtract 7
$ 6x < 30 $

Step 2: Divide by 6
$ x < 5 $

✔ Solution: $ x < 5 $
➡️ Graph: Open circle at 5, shaded to the left.

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⑥ $ 6 - 5x > -10 $

Step 1: Subtract 6
$ -5x > -16 $

Step 2: Divide by -5 → Reverse inequality
$ x < \frac{16}{5} $ → $ x < 3.2 $

✔ Solution: $ x < 3.2 $
➡️ Graph: Open circle at 3.2, shaded to the left.

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⑦ $ 7 + 4x \geq 19 $

Step 1: Subtract 7
$ 4x \geq 12 $

Step 2: Divide by 4
$ x \geq 3 $

✔ Solution: $ x \geq 3 $
➡️ Graph: Closed circle at 3, shaded to the right.

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⑧ $ 5y < 8 - 7 $

Step 1: Simplify right side
$ 5y < 1 $

Step 2: Divide by 5
$ y < \frac{1}{5} $

✔ Solution: $ y < \frac{1}{5} $
➡️ Graph: Open circle at $ \frac{1}{5} $, shaded to the left.

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⑨ $ 6x - 7 \leq 5 $

Step 1: Add 7
$ 6x \leq 12 $

Step 2: Divide by 6
$ x \leq 2 $

✔ Solution: $ x \leq 2 $
➡️ Graph: Closed circle at 2, shaded to the left.

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⑩ $ 6x + 2 \leq 32 $

Step 1: Subtract 2
$ 6x \leq 30 $

Step 2: Divide by 6
$ x \leq 5 $

✔ Solution: $ x \leq 5 $
➡️ Graph: Closed circle at 5, shaded to the left.

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✔ Final Answers Summary:

| Problem | Solution |
|--------|----------|
| ① | $ x \geq \frac{1}{2} $ |
| ② | $ x \leq 3 $ |
| ③ | $ x > -2 $ |
| ④ | $ x > 7 $ |
| ⑤ | $ x < 5 $ |
| ⑥ | $ x < 3.2 $ |
| ⑦ | $ x \geq 3 $ |
| ⑧ | $ y < \frac{1}{5} $ |
| ⑨ | $ x \leq 2 $ |
| ⑩ | $ x \leq 5 $ |

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📌 Graphing Tips:

- Use closed circle (●) for $ \leq $ or $ \geq $
- Use open circle (○) for $ < $ or $ > $
- Shade in the direction that satisfies the inequality:
- For $ x > a $: shade right
- For $ x < a $: shade left

Let me know if you'd like a visual representation of any of these graphs!