Compound Inequalities Notes and Worksheets - Lindsay Bowden - Free Printable

Let's solve each of the compound inequalities step by step and explain how to graph them on a number line. We'll go through each problem one at a time.

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1. $-5 \leq 3x + 4 < 13$

This is a compound inequality with "and" (implied between two inequalities).
We solve it by isolating $x$ in the middle.

Step 1: Subtract 4 from all parts:
$$
-5 - 4 \leq 3x + 4 - 4 < 13 - 4
$$
$$
-9 \leq 3x < 9
$$

Step 2: Divide all parts by 3:
$$
\frac{-9}{3} \leq x < \frac{9}{3}
$$
$$
-3 \leq x < 3
$$

✔ Solution: $[-3, 3)$

Graph:
- Closed circle at $-3$ (inclusive)
- Open circle at $3$ (exclusive)
- Shade between them.

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2. The weights of the wrestling team are either below 150 or greater than or equal to 175

This is a compound inequality with "or".

Let $w$ = weight.

So:
$$
w < 150 \quad \text{or} \quad w \geq 175
$$

✔ Solution: $(-\infty, 150) \cup [175, \infty)$

Graph:
- Open circle at 150, shade left
- Closed circle at 175, shade right
- Two separate rays.

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3. $5x < -25$ or $-3x + 1 \leq 4$

Solve each inequality separately.

First inequality:
$$
5x < -25 \Rightarrow x < -5
$$

Second inequality:
$$
-3x + 1 \leq 4
\Rightarrow -3x \leq 3
\Rightarrow x \geq -1 \quad \text{(Divide by -3 → flip inequality)}
$$

So we have:
$$
x < -5 \quad \text{or} \quad x \geq -1
$$

✔ Solution: $(-\infty, -5) \cup [-1, \infty)$

Graph:
- Open circle at $-5$, shade left
- Closed circle at $-1$, shade right
- Two separate rays.

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4. $2 < \frac{x}{3} - 1 \leq 4$

Solve this compound inequality.

Step 1: Add 1 to all parts:
$$
2 + 1 < \frac{x}{3} - 1 + 1 \leq 4 + 1
\Rightarrow 3 < \frac{x}{3} \leq 5
$$

Step 2: Multiply all parts by 3:
$$
3 \cdot 3 < x \leq 5 \cdot 3
\Rightarrow 9 < x \leq 15
$$

✔ Solution: $(9, 15]$

Graph:
- Open circle at 9
- Closed circle at 15
- Shade between.

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5. A doctor tells you to keep your daily calorie intake above 1500 and less than or equal to 2000

Let $c$ = calories.

So:
$$
c > 1500 \quad \text{and} \quad c \leq 2000
$$

Combine:
$$
1500 < c \leq 2000
$$

✔ Solution: $(1500, 2000]$

Graph:
- Open circle at 1500
- Closed circle at 2000
- Shade between.

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6. $\frac{1}{2}x - 2 \leq 3$ or $2x - 3 > 25$

Solve each inequality separately.

First inequality:
$$
\frac{1}{2}x - 2 \leq 3
\Rightarrow \frac{1}{2}x \leq 5
\Rightarrow x \leq 10
$$

Second inequality:
$$
2x - 3 > 25
\Rightarrow 2x > 28
\Rightarrow x > 14
$$

So:
$$
x \leq 10 \quad \text{or} \quad x > 14
$$

✔ Solution: $(-\infty, 10] \cup (14, \infty)$

Graph:
- Closed circle at 10, shade left
- Open circle at 14, shade right
- Two separate rays.

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✔ Summary of All Solutions:

| Problem | Solution | Graph Description |
|--------|----------|-------------------|
| 1 | $[-3, 3)$ | Closed at -3, open at 3, shaded between |
| 2 | $(-\infty, 150) \cup [175, \infty)$ | Open at 150 (left), closed at 175 (right), both sides shaded |
| 3 | $(-\infty, -5) \cup [-1, \infty)$ | Open at -5 (left), closed at -1 (right), both sides shaded |
| 4 | $(9, 15]$ | Open at 9, closed at 15, shaded between |
| 5 | $(1500, 2000]$ | Open at 1500, closed at 2000, shaded between |
| 6 | $(-\infty, 10] \cup (14, \infty)$ | Closed at 10 (left), open at 14 (right), both sides shaded |

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You can now fill in the number lines accordingly using these solutions.

Let me know if you'd like me to draw the graphs or provide a visual description!