Algebra 1 Worksheets | Inequalities Worksheets - 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:
- "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:
$$
x < 2
$$

---

#### ✔ Solve $ 2x - 1 \geq 5 $:

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

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

---

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

This means any number less than 2 OR greater than or equal to 3 satisfies the compound inequality.

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Graph (for OR):

On a number line:
- Draw an open circle at 2 and shade to the left (for $ x < 2 $).
- Draw a closed circle at 3 and shade to the right (for $ x \geq 3 $).
- The graph will have two separate shaded regions.

```
<---|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|---->
-4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11
o=====================> <======================o
(open at 2) (closed at 3)
```

But since it's "OR", we combine both parts:
- Shade everything left of 2 (not including 2)
- Shade everything right of 3 (including 3)

So the final graph looks like:
```
<---|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|---->
-4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11
o=====================> <======================o
```

✔ So on the blank graph under "OR", you would:
- Put an open circle at 2, shade left.
- Put a closed circle at 3, shade right.
- The middle part (from 2 to 3) is not shaded.

---

## 🔹 Part 2: AND Compound Inequality

Example:

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

This is a compound inequality with "and" — meaning both conditions must be true simultaneously.

We solve this as one inequality by breaking it into two parts:

$$
-2 \leq 3x - 8 \quad \text{AND} \quad 3x - 8 \leq 10
$$

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✔ Step 1: Solve $ -2 \leq 3x - 8 $

Add 8 to both sides:
$$
6 \leq 3x
$$

Divide by 3:
$$
2 \leq x \quad \text{or} \quad x \geq 2
$$

---

✔ Step 2: Solve $ 3x - 8 \leq 10 $

Add 8:
$$
3x \leq 18
$$

Divide by 3:
$$
x \leq 6
$$

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✔ Combine:

$$
x \geq 2 \quad \text{AND} \quad x \leq 6
$$

So the solution is:
$$
2 \leq x \leq 6
$$

---

Graph (for AND):

On the number line:
- Closed circle at 2 (inclusive)
- Closed circle at 6 (inclusive)
- Shade the region between them.

```
<---|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|---->
-4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11
[=====================]
```

✔ So on the blank graph under "AND", draw:
- A closed circle at 2
- A closed circle at 6
- Shade the segment between 2 and 6

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## ✔ Final Summary:

FILL IN THE BLANKS:

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

---

OR Section:

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

- Graph:
Shade left of 2 (open circle), and right of 3 (closed circle). No shading between.

---

AND Section:

- Example:
$$
-2 \leq 3x - 8 \leq 10
$$
Solution: $ 2 \leq x \leq 6 $

- Graph:
Closed circles at 2 and 6, shaded between.

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