Compound Inequalities Notes and Worksheets - Lindsay Bowden - Free Printable

Let's solve the compound inequalities shown in your worksheet and complete the notes. We'll go through each section step by step.

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Definition:

Compound inequality – a set of two or more inequalities that are separated by the word "and" or "or"

✔ Fill in the blanks:
> compound inequality - a set of two or more inequalities that are separated by the word and or or

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## 🔹 Part 1: OR Compound Inequality

Example:

$$
3x + 1 < 7 \quad \text{or} \quad 2x - 1 \geq 5
$$

We solve each inequality separately.

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✔ Solve $ 3x + 1 < 7 $

Subtract 1 from both sides:
$$
3x < 6
$$

Divide by 3:
$$
x < 2
$$

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✔ Solve $ 2x - 1 \geq 5 $

Add 1 to both sides:
$$
2x \geq 6
$$

Divide by 2:
$$
x \geq 3
$$

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✔ Combine with "OR"

The solution is all values where either $ x < 2 $ or $ x \geq 3 $

So, the solution set is:
$$
x < 2 \quad \text{or} \quad x \geq 3
$$

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📈 Graph (on number line):

- Draw a number line.
- For $ x < 2 $: open circle at 2, shade to the left.
- For $ x \geq 3 $: closed circle at 3, shade to the right.
- Since it’s “OR”, we combine both shaded regions.

✔ The graph will have:
- Shading from negative infinity to 2 (open circle at 2)
- And shading from 3 to positive infinity (closed circle at 3)

There is a gap between 2 and 3.

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## 🔹 Part 2: AND Compound Inequality

Example:

$$
-2 \leq 3x - 8 \leq 10
$$

This is a compound inequality with "AND", meaning we want values of $ x $ that satisfy both parts simultaneously.

We can solve this as one continuous inequality.

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✔ Solve $ -2 \leq 3x - 8 \leq 10 $

Add 8 to all three parts:
$$
-2 + 8 \leq 3x \leq 10 + 8
\Rightarrow 6 \leq 3x \leq 18
$$

Divide all parts by 3:
$$
2 \leq x \leq 6
$$

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✔ Solution:

$$
x \in [2, 6]
$$

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📈 Graph (on number line):

- Draw a number line.
- Closed circles at 2 and 6 (since inequalities include equality).
- Shade the segment between 2 and 6.

✔ This represents all numbers from 2 to 6 inclusive.

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## ✔ Final Completed Notes:

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COMPOUND INEQUALITIES notes

> compound inequality – a set of two or more inequalities that are separated by the word and or or

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OR

| JOINING WORD | OR |
|--------------|----|
| EXAMPLE | $ 3x + 1 < 7 $ or $ 2x - 1 \geq 5 $<br>Solution: $ x < 2 $ or $ x \geq 3 $ |
| GRAPH | ![Graph: open circle at 2, shaded left; closed circle at 3, shaded right] |

> *(On number line: shade left of 2 (open) and right of 3 (closed), gap in between)*

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AND

| JOINING WORD | AND |
|--------------|-----|
| EXAMPLE | $ -2 \leq 3x - 8 \leq 10 $<br>Solution: $ 2 \leq x \leq 6 $ |
| GRAPH | ![Graph: closed circles at 2 and 6, shaded between] |

> *(On number line: closed circles at 2 and 6, solid line connecting them)*

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💡 Key Takeaways:

- "OR" means union: solutions from either inequality.
- "AND" means intersection: only values that satisfy both.
- "AND" often appears implicitly in compound inequalities like $ a \leq x \leq b $.

Let me know if you'd like a visual sketch of the graphs!