Math worksheet titled "Solving Inequalities (B)" featuring two sections with algebraic inequalities to solve, including hints and a logo for cazoom!.

Let’s solve each inequality step by step. We’ll go one at a time, carefully isolating the variable x on one side.

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Section A

1) 5x + 4 ≤ 14

Subtract 4 from both sides:
→ 5x ≤ 10
Divide by 5:
→ x ≤ 2

✔ Final Answer for #1: x ≤ 2

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2) 6x – 1 > 23

Add 1 to both sides:
→ 6x > 24
Divide by 6:
→ x > 4

✔ Final Answer for #2: x > 4

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3) (11x + 22)/2 ≤ 33

Multiply both sides by 2:
→ 11x + 22 ≤ 66
Subtract 22:
→ 11x ≤ 44
Divide by 11:
→ x ≤ 4

✔ Final Answer for #3: x ≤ 4

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4) (6x)/8 – 5 < 4

Add 5 to both sides:
→ (6x)/8 < 9
Multiply both sides by 8:
→ 6x < 72
Divide by 6:
→ x < 12

✔ Final Answer for #4: x < 12

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5) 3(4x + 2) > 9

Distribute the 3:
→ 12x + 6 > 9
Subtract 6:
→ 12x > 3
Divide by 12:
→ x > 3/12 → simplify to x > 1/4

✔ Final Answer for #5: x > 1/4

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6) [2(2 + 5x)] / 8 ≤ 3

First, multiply numerator: 2*(2+5x) = 4 + 10x
So: (4 + 10x)/8 ≤ 3
Multiply both sides by 8:
→ 4 + 10x ≤ 24
Subtract 4:
→ 10x ≤ 20
Divide by 10:
→ x ≤ 2

✔ Final Answer for #6: x ≤ 2

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7) 6x + 15 ≥ 9x

Subtract 6x from both sides:
→ 15 ≥ 3x
Divide by 3:
→ 5 ≥ x → or x ≤ 5

✔ Final Answer for #7: x ≤ 5

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8) 5x – 11 > 3x + 9

Subtract 3x from both sides:
→ 2x – 11 > 9
Add 11:
→ 2x > 20
Divide by 2:
→ x > 10

✔ Final Answer for #8: x > 10

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9) 4(3x + 5) < 5(2x + 5)

Expand both sides:
Left: 12x + 20
Right: 10x + 25
So: 12x + 20 < 10x + 25
Subtract 10x:
→ 2x + 20 < 25
Subtract 20:
→ 2x < 5
Divide by 2:
→ x < 5/2 → or x < 2.5

✔ Final Answer for #9: x < 2.5

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10) (7x + 4)/3 ≤ x + 12

Multiply both sides by 3:
→ 7x + 4 ≤ 3x + 36
Subtract 3x:
→ 4x + 4 ≤ 36
Subtract 4:
→ 4x ≤ 32
Divide by 4:
→ x ≤ 8

✔ Final Answer for #10: x ≤ 8

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11) 4 ≤ x + 2 < 17

This is a compound inequality. Solve separately as hinted.

Left part: 4 ≤ x + 2 → subtract 2 → 2 ≤ x
Right part: x + 2 < 17 → subtract 2 → x < 15
Combine: 2 ≤ x < 15

✔ Final Answer for #11: 2 ≤ x < 15

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12) 6 ≤ [3(2x – 5)] / 10 ≤ 12

Multiply all parts by 10:
→ 60 ≤ 3(2x – 5) ≤ 120
Divide all parts by 3:
→ 20 ≤ 2x – 5 ≤ 40
Add 5 to all parts:
→ 25 ≤ 2x ≤ 45
Divide by 2:
→ 12.5 ≤ x ≤ 22.5

✔ Final Answer for #12: 12.5 ≤ x ≤ 22.5

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Section B

1) 7 – 3x > 25

Subtract 7:
→ –3x > 18
Divide by –3 → remember to flip inequality sign!
→ x < –6

✔ Final Answer for #1: x < –6

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2) 1 – 5x ≤ 41

Subtract 1:
→ –5x ≤ 40
Divide by –5 → flip sign!
→ x ≥ –8

✔ Final Answer for #2: x ≥ –8

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3) 2 – x/5 ≥ 8

Subtract 2:
→ –x/5 ≥ 6
Multiply both sides by 5:
→ –x ≥ 30
Multiply by –1 → flip sign!
→ x ≤ –30

✔ Final Answer for #3: x ≤ –30

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4) 3 – 7x ≤ –11

Subtract 3:
→ –7x ≤ –14
Divide by –7 → flip sign!
→ x ≥ 2

✔ Final Answer for #4: x ≥ 2

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5) 9x – 7 < –34

Add 7:
→ 9x < –27
Divide by 9:
→ x < –3

✔ Final Answer for #5: x < –3

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6) [2(6 – 2x)] / 4 ≤ 12

Simplify numerator: 2*(6–2x) = 12 – 4x
So: (12 – 4x)/4 ≤ 12
Which simplifies to: 3 – x ≤ 12
Subtract 3:
→ –x ≤ 9
Multiply by –1 → flip sign!
→ x ≥ –9

✔ Final Answer for #6: x ≥ –9

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7) 7x – 7 < 5x + 15

Subtract 5x:
→ 2x – 7 < 15
Add 7:
→ 2x < 22
Divide by 2:
→ x < 11

✔ Final Answer for #7: x < 11

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8) –12x – 3 ≤ –2x + 47

Add 12x to both sides:
→ –3 ≤ 10x + 47
Subtract 47:
→ –50 ≤ 10x
Divide by 10:
→ –5 ≤ x → or x ≥ –5

✔ Final Answer for #8: x ≥ –5

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9) 9(6 – 2x) > 2(2 – 7x)

Expand both sides:
Left: 54 – 18x
Right: 4 – 14x
So: 54 – 18x > 4 – 14x
Add 18x to both sides:
→ 54 > 4 + 4x
Subtract 4:
→ 50 > 4x
Divide by 4:
→ 12.5 > x → or x < 12.5

✔ Final Answer for #9: x < 12.5

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10) –5 ≤ –2x < 10

Break into two parts:

Left: –5 ≤ –2x → divide by –2 → flip sign → 2.5 ≥ x → x ≤ 2.5
Right: –2x < 10 → divide by –2 → flip sign → x > –5
Combine: –5 < x ≤ 2.5

Wait — let’s do it together without breaking:

Start with: –5 ≤ –2x < 10
Divide ALL parts by –2 → FLIP BOTH SIGNS:
→ 2.5 ≥ x > –5 → which is same as –5 < x ≤ 2.5

✔ Final Answer for #10: –5 < x ≤ 2.5

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11) 4x – 10 ≤ 2(x – 1) < 8 + 3x

This is a compound inequality. Break into two parts.

Part 1: 4x – 10 ≤ 2(x – 1)
→ 4x – 10 ≤ 2x – 2
Subtract 2x:
→ 2x – 10 ≤ –2
Add 10:
→ 2x ≤ 8
→ x ≤ 4

Part 2: 2(x – 1) < 8 + 3x
→ 2x – 2 < 8 + 3x
Subtract 2x:
→ –2 < 8 + x
Subtract 8:
→ –10 < x → or x > –10

Combine: –10 < x ≤ 4

✔ Final Answer for #11: –10 < x ≤ 4

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12) –12 ≤ [4(2 – x)] / 3 ≤ (3x – 6)/4

This one has different denominators. Let’s handle left and right inequalities separately.

First, write clearly:

Inequality A: –12 ≤ [4(2 – x)] / 3
Inequality B: [4(2 – x)] / 3 ≤ (3x – 6)/4

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Solve Inequality A: –12 ≤ [4(2 – x)] / 3

Multiply both sides by 3:
→ –36 ≤ 4(2 – x)
→ –36 ≤ 8 – 4x
Subtract 8:
→ –44 ≤ –4x
Divide by –4 → flip sign:
→ 11 ≥ x → or x ≤ 11

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Solve Inequality B: [4(2 – x)] / 3 ≤ (3x – 6)/4

Cross-multiply (since denominators are positive, no sign flip):
→ 4 * 4(2 – x) ≤ 3 * (3x – 6)
→ 16(2 – x) ≤ 9x – 18
→ 32 – 16x ≤ 9x – 18
Add 16x to both sides:
→ 32 ≤ 25x – 18
Add 18:
→ 50 ≤ 25x
Divide by 25:
→ 2 ≤ x → or x ≥ 2

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Now combine both results:
From A: x ≤ 11
From B: x ≥ 2
So: 2 ≤ x ≤ 11

✔ Final Answer for #12: 2 ≤ x ≤ 11

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## ✔ FINAL ANSWERS (All Problems)

Section A

1) x ≤ 2
2) x > 4
3) x ≤ 4
4) x < 12
5) x > 1/4
6) x ≤ 2
7) x ≤ 5
8) x > 10
9) x < 2.5
10) x ≤ 8
11) 2 ≤ x < 15
12) 12.5 ≤ x ≤ 22.5

Section B

1) x < –6
2) x ≥ –8
3) x ≤ –30
4) x ≥ 2
5) x < –3
6) x ≥ –9
7) x < 11
8) x ≥ –5
9) x < 12.5
10) –5 < x ≤ 2.5
11) –10 < x ≤ 4
12) 2 ≤ x ≤ 11