Worksheet for solving multi-step inequalities with graphing exercises.

Let’s solve each inequality step by step.

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Problem ①: 5y + 8y < 13

Step 1: Combine like terms on the left.
→ 13y < 13

Step 2: Divide both sides by 13.
→ y < 1

Graph: Open circle at 1, shade to the left.

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Problem ②: 9 ≥ -2p + 2 - 3

Step 1: Simplify right side: 2 - 3 = -1
→ 9 ≥ -2p - 1

Step 2: Add 1 to both sides.
→ 10 ≥ -2p

Step 3: Divide both sides by -2. Remember: dividing by negative flips the inequality!
→ -5 ≤ p (or p ≥ -5)

Graph: Closed circle at -5, shade to the right.

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Problem ③: -9 > -5q - 4q

Step 1: Combine like terms on right.
→ -9 > -9q

Step 2: Divide both sides by -9. Flip inequality!
→ 1 < q (or q > 1)

Graph: Open circle at 1, shade to the right.

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Problem ④: -14 ≤ 7 + 7n - 7

Step 1: Simplify right side: 7 - 7 = 0
→ -14 ≤ 7n

Step 2: Divide both sides by 7.
→ -2 ≤ n (or n ≥ -2)

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

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Problem ⑤: 5(6 + 3t) + 7 ≥ 127

Step 1: Distribute the 5.
→ 30 + 15t + 7 ≥ 127

Step 2: Combine constants.
→ 37 + 15t ≥ 127

Step 3: Subtract 37 from both sides.
→ 15t ≥ 90

Step 4: Divide by 15.
→ t ≥ 6

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

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Problem ⑥: 13 + 2v - 8 + 6 > -7 - v

Step 1: Combine constants on left: 13 - 8 + 6 = 11
→ 11 + 2v > -7 - v

Step 2: Add v to both sides.
→ 11 + 3v > -7

Step 3: Subtract 11 from both sides.
→ 3v > -18

Step 4: Divide by 3.
→ v > -6

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

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Problem ⑦: -24 ≤ 6(5b - 2) - 8(8b - 7)

Step 1: Distribute both parts.
→ -24 ≤ 30b - 12 - 64b + 56

Step 2: Combine like terms on right.
30b - 64b = -34b
-12 + 56 = 44
→ -24 ≤ -34b + 44

Step 3: Subtract 44 from both sides.
→ -68 ≤ -34b

Step 4: Divide by -34. Flip inequality!
→ 2 ≥ b (or b ≤ 2)

Graph: Closed circle at 2, shade to the left.

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Problem ⑧: x - 8 + 3x + 2 < -6(8x - 7) + 4(8x - 2)

Step 1: Combine like terms on left.
x + 3x = 4x
-8 + 2 = -6
→ 4x - 6 < ...

Step 2: Distribute on right.
-6(8x - 7) = -48x + 42
4(8x - 2) = 32x - 8
→ 4x - 6 < -48x + 42 + 32x - 8

Step 3: Combine like terms on right.
-48x + 32x = -16x
42 - 8 = 34
→ 4x - 6 < -16x + 34

Step 4: Add 16x to both sides.
→ 20x - 6 < 34

Step 5: Add 6 to both sides.
→ 20x < 40

Step 6: Divide by 20.
→ x < 2

Graph: Open circle at 2, shade to the left.

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Final Answer:
① y < 1
② p ≥ -5
③ q > 1
④ n ≥ -2
⑤ t ≥ 6
⑥ v > -6
⑦ b ≤ 2
⑧ x < 2