Two-Step Inequalities.ks-ipa - Kuta Software - Free Printable

Let's solve each of the two-step inequalities step by step and then graph the solution on the number line.

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1) $\frac{m}{3} + 2 > 0$

Step 1: Subtract 2 from both sides:
$$
\frac{m}{3} > -2
$$

Step 2: Multiply both sides by 3:
$$
m > -6
$$

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

✔ Solution: $m > -6$

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2) $\frac{p}{9} - 1 \leq -2$

Step 1: Add 1 to both sides:
$$
\frac{p}{9} \leq -1
$$

Step 2: Multiply both sides by 9:
$$
p \leq -9
$$

Graph: Closed circle at $-9$, shade to the left.

✔ Solution: $p \leq -9$

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3) $\frac{z}{7} + 5 > 5$

Step 1: Subtract 5 from both sides:
$$
\frac{z}{7} > 0
$$

Step 2: Multiply both sides by 7:
$$
z > 0
$$

Graph: Open circle at $0$, shade to the right.

✔ Solution: $z > 0$

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4) $\frac{1 + m}{9} \geq 1$

Step 1: Multiply both sides by 9:
$$
1 + m \geq 9
$$

Step 2: Subtract 1 from both sides:
$$
m \geq 8
$$

Graph: Closed circle at $8$, shade to the right.

✔ Solution: $m \geq 8$

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5) $-3r - 2 \leq 4$

Step 1: Add 2 to both sides:
$$
-3r \leq 6
$$

Step 2: Divide both sides by $-3$ (remember to reverse the inequality when dividing by a negative):
$$
r \geq -2
$$

Graph: Closed circle at $-2$, shade to the right.

✔ Solution: $r \geq -2$

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6) $8x + 2 \leq 138$

Step 1: Subtract 2 from both sides:
$$
8x \leq 136
$$

Step 2: Divide by 8:
$$
x \leq 17
$$

Graph: Closed circle at $17$, shade to the left.

✔ Solution: $x \leq 17$

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7) $3 - \frac{b}{9} < 4$

Step 1: Subtract 3 from both sides:
$$
-\frac{b}{9} < 1
$$

Step 2: Multiply both sides by $-9$ (reverse inequality):
$$
b > -9
$$

Graph: Open circle at $-9$, shade to the right.

✔ Solution: $b > -9$

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8) $9 + \frac{n}{2} > 16$

Step 1: Subtract 9 from both sides:
$$
\frac{n}{2} > 7
$$

Step 2: Multiply both sides by 2:
$$
n > 14
$$

Graph: Open circle at $14$, shade to the right.

✔ Solution: $n > 14$

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✔ Final Answers:

| Problem | Solution |
|--------|----------|
| 1) | $m > -6$ |
| 2) | $p \leq -9$ |
| 3) | $z > 0$ |
| 4) | $m \geq 8$ |
| 5) | $r \geq -2$ |
| 6) | $x \leq 17$ |
| 7) | $b > -9$ |
| 8) | $n > 14$ |

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📌 Graphing Tips:

- Use an open circle for $<$ or $>$.
- Use a closed circle for $\leq$ or $\geq$.
- Shade in the direction that satisfies the inequality.

Let me know if you'd like visual graphs drawn!