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 step-by-step for both OR and AND cases.

---

🔹 Definition:

> A compound inequality is a set of two or more inequalities that are separated by the word "and" or "or".

So, fill in:
> compound inequality - a set of two or more inequalities that are separated by the word and or or

---

## ✔ Part 1: OR Inequality

Example:

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

We solve each inequality separately.

---

Step 1: Solve $3x + 1 < 7$

$$
3x + 1 < 7 \\
3x < 6 \\
x < 2
$$

---

Step 2: Solve $2x - 1 \geq 5$

$$
2x - 1 \geq 5 \\
2x \geq 6 \\
x \geq 3
$$

---

Solution for "OR":

The solution is all values that satisfy either inequality.

So:
$$
x < 2 \quad \text{or} \quad x \geq 3
$$

This means any number less than 2 or greater than or equal to 3.

---

Graph:

On a number line:

- Draw an open circle at 2 (since it's not included) and shade to the left.
- Draw a closed circle at 3 (since it's included) and shade to the right.

```
<----●------------------○------>
2 3
```

But since it's OR, we combine both parts:

```
<---●=================○-------->
2 3
```

Wait — actually, this is two separate regions:

- Shade from $-\infty$ to 2 (not including 2)
- Shade from 3 to $\infty$ (including 3)

So the graph looks like:

```
<---●=================○-------->
2 3
```

But with a gap between 2 and 3.

Correct graph:

```
<---●================= ==========○-------->
2 3
```

So:
- Open circle at 2 → shaded left
- Closed circle at 3 → shaded right
- Gap between 2 and 3

✔ So, the graph should show:
- An open circle at 2, arrow going left
- A closed circle at 3, arrow going right
- No shading between 2 and 3

---

## ✔ Part 2: AND 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 at the same time.

We can solve it in one step by doing operations to all three parts.

---

Step: Solve $-2 \leq 3x - 8 \leq 10$

Add 8 to all parts:

$$
-2 + 8 \leq 3x - 8 + 8 \leq 10 + 8 \\
6 \leq 3x \leq 18
$$

Now divide all parts by 3:

$$
\frac{6}{3} \leq x \leq \frac{18}{3} \\
2 \leq x \leq 6
$$

---

Solution:

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

All numbers from 2 to 6, inclusive.

---

Graph:

- Closed circle at 2
- Closed circle at 6
- Shaded line between them

```
●===================●
2 6
```

---

## ✔ Final Completed Notes

Compound Inequality Definition:

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

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OR Section

| JOINING WORD | OR |
|--------------|----|
| EXAMPLE | $3x + 1 < 7$ or $2x - 1 \geq 5$ → $x < 2$ or $x \geq 3$ |
| GRAPH |  |

> *(Draw: open circle at 2, arrow left; closed circle at 3, arrow right)*

---

AND Section

| JOINING WORD | AND |
|--------------|-----|
| EXAMPLE | $-2 \leq 3x - 8 \leq 10$ → $2 \leq x \leq 6$ |
| GRAPH |  |

> *(Draw: closed circle at 2, closed circle at 6, solid line between them)*

---

Summary:

- OR: Union of solutions (either one works)
- AND: Intersection of solutions (must satisfy both)

Let me know if you'd like this as a printable version!