Multi-Step Inequalities Worksheet - Free Printable

Let's solve each of the multi-step inequalities step by step, and verify the solutions given in the worksheet. We'll go through each problem, show the algebraic steps, and explain how to graph the solution on a number line.

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1) $ 3 < -5n + 2n $

Step 1: Combine like terms on the right-hand side:
$$
3 < -3n
$$

Step 2: Divide both sides by $-3$. Remember: when dividing by a negative number, reverse the inequality sign.
$$
\frac{3}{-3} > n \quad \Rightarrow \quad -1 > n \quad \text{or} \quad n < -1
$$

✔ Solution: $ n < -1 $
✔️ Graph: Open circle at $-1$, shaded to the left → Correct.

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2) $ 6x + 2 + 6x < 14 $

Step 1: Combine like terms:
$$
12x + 2 < 14
$$

Step 2: Subtract 2 from both sides:
$$
12x < 12
$$

Step 3: Divide by 12:
$$
x < 1
$$

✔ Solution: $ x < 1 $
✔️ Graph: Open circle at 1, shaded left → Correct.

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3) $ -p - 4p > -10 $

Step 1: Combine like terms:
$$
-5p > -10
$$

Step 2: Divide by $-5$ (reverse inequality):
$$
p < 2
$$

✔ Solution: $ p < 2 $
✔️ Graph: Open circle at 2, shaded left → Correct.

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4) $ 18 \geq 5k + 4k $

Step 1: Combine like terms:
$$
18 \geq 9k
$$

Step 2: Divide by 9:
$$
2 \geq k \quad \text{or} \quad k \leq 2
$$

✔ Solution: $ k \leq 2 $
✔️ Graph: Closed circle at 2, shaded left → Correct.

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5) $ 9 \geq -2m + 2 - 3 $

Step 1: Simplify right-hand side:
$$
9 \geq -2m -1
$$

Step 2: Add 1 to both sides:
$$
10 \geq -2m
$$

Step 3: Divide by $-2$ (reverse inequality):
$$
-5 \leq m \quad \text{or} \quad m \geq -5
$$

✔ Solution: $ m \geq -5 $
✔️ Graph: Closed circle at $-5$, shaded right → Correct.

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6) $ -3 - 6(4x + 6) > -111 $

Step 1: Distribute:
$$
-3 -24x -36 > -111
$$

Step 2: Combine constants:
$$
-24x -39 > -111
$$

Step 3: Add 39 to both sides:
$$
-24x > -72
$$

Step 4: Divide by $-24$ (reverse inequality):
$$
x < 3
$$

✔ Solution: $ x < 3 $
✔️ Graph: Open circle at 3, shaded left → Correct.

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7) $ 6 - 4(6n + 7) \geq 122 $

Step 1: Distribute:
$$
6 -24n -28 \geq 122
$$

Step 2: Combine constants:
$$
-24n -22 \geq 122
$$

Step 3: Add 22 to both sides:
$$
-24n \geq 144
$$

Step 4: Divide by $-24$ (reverse inequality):
$$
n \leq -6
$$

✔ Solution: $ n \leq -6 $
✔️ Graph: Closed circle at $-6$, shaded left → Correct.

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8) $ -138 \geq -6(6b - 7) $

Step 1: Distribute:
$$
-138 \geq -36b + 42
$$

Step 2: Subtract 42 from both sides:
$$
-180 \geq -36b
$$

Step 3: Divide by $-36$ (reverse inequality):
$$
5 \leq b \quad \text{or} \quad b \geq 5
$$

✔ Solution: $ b \geq 5 $
✔️ Graph: Closed circle at 5, shaded right → Correct.

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9) $ 167 < 6 + 7(2 - 7r) $

Step 1: Distribute:
$$
167 < 6 + 14 - 49r
$$

Step 2: Combine:
$$
167 < 20 - 49r
$$

Step 3: Subtract 20:
$$
147 < -49r
$$

Step 4: Divide by $-49$ (reverse inequality):
$$
-3 > r \quad \text{or} \quad r < -3
$$

✔ Solution: $ r < -3 $
✔️ Graph: Open circle at $-3$, shaded left → Correct.

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10) $ 5(6 + 3r) + 7 \geq 127 $

Step 1: Distribute:
$$
30 + 15r + 7 \geq 127
$$

Step 2: Combine:
$$
15r + 37 \geq 127
$$

Step 3: Subtract 37:
$$
15r \geq 90
$$

Step 4: Divide by 15:
$$
r \geq 6
$$

✔ Solution: $ r \geq 6 $
✔️ Graph: Closed circle at 6, shaded right → Correct.

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11) $ -8x + 2x - 16 < -5x + 7x $

Step 1: Combine like terms on both sides:
Left: $ -6x - 16 $
Right: $ 2x $

So:
$$
-6x - 16 < 2x
$$

Step 2: Subtract $2x$ from both sides:
$$
-8x - 16 < 0
$$

Step 3: Add 16:
$$
-8x < 16
$$

Step 4: Divide by $-8$ (reverse inequality):
$$
x > -2
$$

✔ Solution: $ x > -2 $
✔️ Graph: Open circle at $-2$, shaded right → Correct.

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12) $ -1 - 6x - 6 > -11 - 7x $

Step 1: Combine like terms:
Left: $ -6x -7 $
Right: $ -11 -7x $

So:
$$
-6x -7 > -11 -7x
$$

Step 2: Add $7x$ to both sides:
$$
x -7 > -11
$$

Step 3: Add 7:
$$
x > -4
$$

✔ Solution: $ x > -4 $
✔️ Graph: Open circle at $-4$, shaded right → Correct.

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✔ Final Summary:

All 12 problems have been solved correctly. The solutions provided in red are accurate, and the graphs match the correct intervals.

Each inequality was solved using standard algebraic techniques:
- Combine like terms
- Use inverse operations
- Isolate the variable
- Reverse inequality when multiplying/dividing by a negative

Graphing rules:
- Open circle: strict inequality ($<$ or $>$)
- Closed circle: inclusive inequality ($\leq$ or $\geq$)
- Shade in the direction that satisfies the inequality

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✔ All answers are correct!