Pre-Algebra Worksheets | Inequalities Worksheets - Free Printable

Let's solve each inequality step by step and explain the solution process. We'll also determine how to graph each solution on the number line provided.

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1) $\frac{x}{3} \geq 5$

Step 1: Multiply both sides by 3 to eliminate the denominator:
$$
x \geq 15
$$

Graph: Draw a closed circle at 15 (since it's "greater than or equal to") and shade to the right.

✔ Answer: $x \geq 15$

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2) $-13 < \frac{y}{2}$

Step 1: Multiply both sides by 2:
$$
-26 < y \quad \text{or} \quad y > -26
$$

Graph: Open circle at -26 (since it's strictly greater), shade to the right.

✔ Answer: $y > -26$

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3) $-4b \geq 28$

Step 1: Divide both sides by -4. Remember: When dividing by a negative number, reverse the inequality sign.
$$
b \leq -7
$$

Graph: Closed circle at -7, shade to the left.

✔ Answer: $b \leq -7$

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4) $52 \geq 4d$

Step 1: Divide both sides by 4:
$$
13 \geq d \quad \text{or} \quad d \leq 13
$$

Graph: Closed circle at 13, shade to the left.

✔ Answer: $d \leq 13$

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5) $\frac{h}{3} < 10$

Step 1: Multiply both sides by 3:
$$
h < 30
$$

Graph: Open circle at 30, shade to the left.

✔ Answer: $h < 30$

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6) $-8 > \frac{z}{7}$

Step 1: Multiply both sides by 7:
$$
-56 > z \quad \text{or} \quad z < -56
$$

Graph: Open circle at -56, shade to the left.

✔ Answer: $z < -56$

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7) $7n > 42$

Step 1: Divide both sides by 7:
$$
n > 6
$$

Graph: Open circle at 6, shade to the right.

✔ Answer: $n > 6$

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8) $91 \leq -7q$

Step 1: Divide both sides by -7. Reverse inequality sign:
$$
-13 \geq q \quad \text{or} \quad q \leq -13
$$

Graph: Closed circle at -13, shade to the left.

✔ Answer: $q \leq -13$

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9) $\frac{v}{4} \geq -8$

Step 1: Multiply both sides by 4:
$$
v \geq -32
$$

Graph: Closed circle at -32, shade to the right.

✔ Answer: $v \geq -32$

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10) $-11 < \frac{k}{4}$

Step 1: Multiply both sides by 4:
$$
-44 < k \quad \text{or} \quad k > -44
$$

Graph: Open circle at -44, shade to the right.

✔ Answer: $k > -44$

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✔ Final Answers Summary:

| Problem | Solution |
|--------|----------|
| 1) | $x \geq 15$ |
| 2) | $y > -26$ |
| 3) | $b \leq -7$ |
| 4) | $d \leq 13$ |
| 5) | $h < 30$ |
| 6) | $z < -56$ |
| 7) | $n > 6$ |
| 8) | $q \leq -13$ |
| 9) | $v \geq -32$ |
| 10) | $k > -44$ |

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🔍 Key Tips:

- Multiplying/dividing by a negative number? Flip the inequality sign.
- "≥" or "≤"? Use a closed circle.
- ">" or "<"? Use an open circle.
- Always shade in the direction that satisfies the inequality.

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