Printable worksheet for solving compound inequalities with ten practice problems.

I'm ready to help you solve the compound inequalities in your worksheet, but I can't see the image you uploaded. However, based on the text you've provided, I can solve each of the compound inequalities listed. Let's go through them one by one.

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1) \( x + 3 > 10 \) and \( x - 8 < 18 \)

Step 1: Solve each inequality separately.

- \( x + 3 > 10 \)
Subtract 3:
\( x > 7 \)

- \( x - 8 < 18 \)
Add 8:
\( x < 26 \)

Step 2: Combine the two results using "and" (intersection).

So, \( x > 7 \) and \( x < 26 \)

Answer: \( 7 < x < 26 \)

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2) \( 2x + 3 < -15 \) or \( 4x > 12 \)

Step 1: Solve each inequality.

- \( 2x + 3 < -15 \)
Subtract 3:
\( 2x < -18 \)
Divide by 2:
\( x < -9 \)

- \( 4x > 12 \)
Divide by 4:
\( x > 3 \)

Step 2: Combine with "or" (union).

So, \( x < -9 \) or \( x > 3 \)

Answer: \( x < -9 \) or \( x > 3 \)

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3) \( \frac{3}{4}x \leq \frac{3}{2} - 3 \)

Wait — this is a single inequality, not compound. Let’s solve it:

\( \frac{3}{4}x \leq \frac{3}{2} - 3 \)

First simplify right side:

\( \frac{3}{2} - 3 = \frac{3}{2} - \frac{6}{2} = -\frac{3}{2} \)

So:
\( \frac{3}{4}x \leq -\frac{3}{2} \)

Multiply both sides by \( \frac{4}{3} \):

\( x \leq -\frac{3}{2} \cdot \frac{4}{3} = -2 \)

Answer: \( x \leq -2 \)

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4) \( 2x + 3 < 10 \) or \( 3x - 1 < 2 \)

Solve each:

- \( 2x + 3 < 10 \) → \( 2x < 7 \) → \( x < 3.5 \)

- \( 3x - 1 < 2 \) → \( 3x < 3 \) → \( x < 1 \)

Now combine with "or":
\( x < 3.5 \) or \( x < 1 \)

But since \( x < 1 \) is already included in \( x < 3.5 \), the union is just \( x < 3.5 \)

Answer: \( x < 3.5 \) or \( x < \frac{7}{2} \)

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5) \( \frac{2}{3}x - 10 < 4 \) or \( -10x \geq 50 \)

Solve each:

- \( \frac{2}{3}x - 10 < 4 \)
Add 10: \( \frac{2}{3}x < 14 \)
Multiply by \( \frac{3}{2} \):
\( x < 14 \cdot \frac{3}{2} = 21 \)

- \( -10x \geq 50 \)
Divide by -10 → reverse inequality:
\( x \leq -5 \)

Now combine with "or":
\( x < 21 \) or \( x \leq -5 \)

Since \( x \leq -5 \) is part of \( x < 21 \), the union is just \( x < 21 \)

Answer: \( x < 21 \)

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6) \( 8x + 24 > -12 \)

This is a single inequality.

\( 8x + 24 > -12 \)
Subtract 24:
\( 8x > -36 \)
Divide by 8:
\( x > -\frac{36}{8} = -4.5 \)

Answer: \( x > -4.5 \) or \( x > -\frac{9}{2} \)

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7) \( -10x + 15 < 4 \) or \( x \geq 10 \)

Solve first inequality:

- \( -10x + 15 < 4 \)
Subtract 15: \( -10x < -11 \)
Divide by -10 → reverse inequality:
\( x > \frac{11}{10} = 1.1 \)

Second: \( x \geq 10 \)

Now combine with "or":
\( x > 1.1 \) or \( x \geq 10 \)

But since \( x \geq 10 \) is already within \( x > 1.1 \), the union is simply \( x > 1.1 \)

Answer: \( x > 1.1 \) or \( x > \frac{11}{10} \)

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8) \( 2(8 - x) \geq 20 \) and \( 8x < -40 \)

Solve each:

- \( 2(8 - x) \geq 20 \)
Distribute: \( 16 - 2x \geq 20 \)
Subtract 16: \( -2x \geq 4 \)
Divide by -2 → reverse inequality:
\( x \leq -2 \)

- \( 8x < -40 \)
Divide by 8: \( x < -5 \)

Now combine with "and" → intersection:

We need both:
\( x \leq -2 \) and \( x < -5 \)

The more restrictive condition is \( x < -5 \), since any number less than -5 is also ≤ -2.

So intersection is \( x < -5 \)

Answer: \( x < -5 \)

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9) \( \frac{1}{2}x + 6 \geq 4 \) and \( -22 > -2x \)

Solve each:

- \( \frac{1}{2}x + 6 \geq 4 \)
Subtract 6: \( \frac{1}{2}x \geq -2 \)
Multiply by 2: \( x \geq -4 \)

- \( -22 > -2x \)
Divide by -2 → reverse inequality:
\( 11 < x \) → \( x > 11 \)

Now combine with "and":
\( x \geq -4 \) and \( x > 11 \)

Intersection: numbers greater than 11 are also ≥ -4, so result is \( x > 11 \)

Answer: \( x > 11 \)

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10) \( -8x + 10 \leq \frac{1}{2}x - 2 \)

This is a single inequality.

Bring all terms to one side:

\( -8x - \frac{1}{2}x \leq -2 - 10 \)

\( -\frac{17}{2}x \leq -12 \)

Multiply both sides by \( -\frac{2}{17} \), reverse inequality:

\( x \geq \frac{-12 \cdot (-2)}{17} = \frac{24}{17} \)

Answer: \( x \geq \frac{24}{17} \)

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✔ Final Answers Summary:

1. \( 7 < x < 26 \)
2. \( x < -9 \) or \( x > 3 \)
3. \( x \leq -2 \)
4. \( x < 3.5 \)
5. \( x < 21 \)
6. \( x > -4.5 \)
7. \( x > 1.1 \)
8. \( x < -5 \)
9. \( x > 11 \)
10. \( x \geq \frac{24}{17} \)

Let me know if you'd like these graphed or explained further!