Inequalities Worksheet for solving algebraic inequalities.

Let’s solve each inequality step by step. I’ll go one at a time, showing all the work so you can follow along easily.

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Problem 1: 6 - b < 8 - 4b

Step 1: Get all the variable terms on one side. 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 4 and the -2:

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) ≤ 5
→ 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 solve it by subtracting 2 from all three 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:

y + 15 ≤ 2

Step 3: Subtract 15 from both sides:

y ≤ -13

✔ Final answer for #4: y ≤ -13

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Problem 5: -5(u - 19) ≤ -6 + 2u

Step 1: Distribute the -5:

-5u + 95 ≤ -6 + 2u

Step 2: Subtract 2u from both sides:

-7u + 95 ≤ -6

Step 3: Subtract 95 from both sides:

-7u ≤ -101

Step 4: Divide by -7 → Remember! When dividing by a negative, flip the inequality sign!

u ≥ 101/7

✔ 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
→ 11 - 2n > -1

Step 2: Subtract 11 from both sides:

-2n > -12

Step 3: Divide by -2 → Flip the inequality!

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. First, find the roots: x = 5 and x = 2.

The expression changes sign at these points. Test intervals:

- For x < 2 (say x=0): (0-5)(0-2) = (-5)(-2) = 10 → positive → not ≤ 0
- For 2 < x < 5 (say x=3): (3-5)(3-2) = (-2)(1) = -2 → negative → satisfies ≤ 0
- For x > 5 (say x=6): (6-5)(6-2) = (1)(4) = 4 → positive → not ≤ 0

Also, since it’s ≤ 0, include the endpoints where it equals zero.

So solution: 2 ≤ 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:

x(x + 7) < 0

Roots are x = 0 and x = -7.

Test intervals:

- x < -7 (say x=-8): (-8)(-1) = 8 → positive → no
- -7 < x < 0 (say x=-1): (-1)(6) = -6 → negative → yes
- x > 0 (say x=1): (1)(8) = 8 → positive → no

Since it’s < 0, 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:

x² + 6 ≥ 106

Step 2: Subtract 6:

x² ≥ 100

Step 3: Take square root → remember, this gives two cases:

x ≥ 10 OR x ≤ -10

✔ Final answer for #10: x ≤ -10 or x ≥ 10

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Final Answers:

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