Printable inequalities worksheet featuring ten problems for solving algebraic inequalities, designed for math practice.

Let’s solve each inequality one by one, step by step. I’ll show all the work so you can follow along easily.

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Problem ①: 6 - b < 8 - 4b

Step 1: Get all the “b” terms on one side and numbers on the other.
Add 4b to both sides:
6 - b + 4b < 8 - 4b + 4b → 6 + 3b < 8

Step 2: Subtract 6 from both sides:
6 + 3b - 6 < 8 - 6 → 3b < 2

Step 3: Divide both sides by 3:
b < 2/3

✔ Final Answer for ①: b < 2/3

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Problem ②: 4(x + 1/2) - 2(x + 3/2) ≤ 5

Step 1: Distribute the numbers outside the parentheses.
4*x + 4*(1/2) = 4x + 2
-2*x -2*(3/2) = -2x - 3
So now: 4x + 2 - 2x - 3 ≤ 5

Step 2: Combine like terms.
(4x - 2x) + (2 - 3) = 2x - 1
So: 2x - 1 ≤ 5

Step 3: Add 1 to both sides:
2x ≤ 6

Step 4: Divide by 2:
x ≤ 3

✔ Final Answer for ②: x ≤ 3

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Problem ③: -1 < x + 2 < 5

This is a compound inequality — we do the same thing to all three parts.

Step 1: Subtract 2 from all parts:
-1 - 2 < x + 2 - 2 < 5 - 2
→ -3 < x < 3

✔ Final Answer for ③: -3 < x < 3

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Problem ④: 3(y + 5) ≤ 2(y + 1)

Step 1: Distribute.
3y + 15 ≤ 2y + 2

Step 2: Subtract 2y from both sides:
3y - 2y + 15 ≤ 2 → y + 15 ≤ 2

Step 3: Subtract 15 from both sides:
y ≤ 2 - 15 → y ≤ -13

✔ Final Answer for ④: y ≤ -13

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Problem ⑤: -5(u - 19) ≤ -6 + 2u

Step 1: Distribute the -5.
-5*u + (-5)*(-19) = -5u + 95
So: -5u + 95 ≤ -6 + 2u

Step 2: Get all u terms on one side. Let’s add 5u to both sides:
95 ≤ -6 + 7u

Step 3: Add 6 to both sides:
95 + 6 ≤ 7u → 101 ≤ 7u

Step 4: Divide by 7:
u ≥ 101/7 → which is about 14.428... but we leave as fraction unless told otherwise.

✔ Final Answer for ⑤: u ≥ 101/7

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Problem ⑥: 3 - 2(n - 4) > -1

Step 1: Distribute the -2.
3 - 2n + 8 > -1 → because -2 * -4 = +8

Step 2: Combine constants: 3 + 8 = 11
So: 11 - 2n > -1

Step 3: Subtract 11 from both sides:
-2n > -12

Step 4: Divide by -2 → REMEMBER: when dividing by negative, flip the inequality sign!
n < 6

✔ Final Answer for ⑥: n < 6

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Problem ⑦: 4r + 3 > 2r + 11

Step 1: Subtract 2r from both sides:
2r + 3 > 11

Step 2: Subtract 3:
2r > 8

Step 3: Divide by 2:
r > 4

✔ Final Answer for ⑦: r > 4

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Problem ⑧: (x - 5)(x - 2) ≤ 0

This is a quadratic inequality. We find where the expression equals zero first.

Set each factor to zero:
x - 5 = 0 → x = 5
x - 2 = 0 → x = 2

These are the critical points. They divide the number line into 3 regions:

Region 1: x < 2 → pick x=0: (0-5)(0-2)=(-5)(-2)=+10 → positive → not ≤ 0
Region 2: 2 < x < 5 → pick x=3: (3-5)(3-2)=(-2)(1)=-2 → negative → YES
Region 3: x > 5 → pick x=6: (6-5)(6-2)=(1)(4)=+4 → positive → no

Also, since it’s ≤ 0, we include the endpoints where it equals zero: x=2 and x=5.

✔ Final Answer for ⑧: 2 ≤ x ≤ 5

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Problem ⑨: x² + 7x + 10 < 10

Step 1: Subtract 10 from both sides:
x² + 7x < 0

Step 2: Factor left side:
x(x + 7) < 0

Find zeros: x = 0, x = -7

Test intervals:

Region 1: x < -7 → pick x=-8: (-8)(-1)=+8 → positive → no
Region 2: -7 < x < 0 → pick x=-1: (-1)(6)=-6 → negative → YES
Region 3: x > 0 → pick x=1: (1)(8)=+8 → positive → no

Since it’s strict inequality (<), we do NOT include endpoints.

✔ Final Answer for ⑨: -7 < x < 0

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Problem ⑩: (x² + 6)/2 ≥ 53

Step 1: Multiply both sides by 2 to eliminate denominator:
x² + 6 ≥ 106

Step 2: Subtract 6:
x² ≥ 100

Step 3: Take square root of both sides → remember: if x² ≥ a, then x ≤ -√a OR x ≥ √a
So: x ≤ -10 or x ≥ 10

✔ Final Answer for ⑩: x ≤ -10 or x ≥ 10

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Final Answer:
① b < 2/3
② x ≤ 3
③ -3 < x < 3
④ y ≤ -13
⑤ u ≥ 101/7
⑥ n < 6
⑦ r > 4
⑧ 2 ≤ x ≤ 5
⑨ -7 < x < 0
⑩ x ≤ -10 or x ≥ 10