A22a - Solving linear inequalities in one variable - BossMaths.com - Free Printable

Let's solve each inequality step by step and represent the solutions on a number line. We'll go through each one carefully.

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1) $-4x < 8$

Step 1: Divide both sides by $-4$.
Remember: When dividing or multiplying both sides of an inequality by a negative number, reverse the inequality sign.

$$
x > \frac{8}{-4} = -2
$$

✔ Solution: $x > -2$

Number Line Representation:
Draw a number line. Place an open circle at $-2$ (since it's not included), and shade to the right.

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2) $3x \leq -7$

Step 1: Divide both sides by 3.

$$
x \leq \frac{-7}{3} \approx -2.333...
$$

✔ Solution: $x \leq -\frac{7}{3}$

Number Line:
Closed circle at $-\frac{7}{3}$, shade to the left.

---

3) $m - 3 < -7$

Step 1: Add 3 to both sides.

$$
m < -7 + 3 = -4
$$

✔ Solution: $m < -4$

Number Line:
Open circle at $-4$, shade to the left.

---

4) $\frac{n+1}{-4} \geq 8$

Step 1: Multiply both sides by $-4$.
Again, since we're multiplying by a negative, reverse the inequality.

$$
n + 1 \leq 8 \times (-4) = -32
$$

Step 2: Subtract 1 from both sides.

$$
n \leq -32 - 1 = -33
$$

✔ Solution: $n \leq -33$

Number Line:
Closed circle at $-33$, shade to the left.

---

5) $7(u + 8) \geq -35$

Step 1: Distribute the 7.

$$
7u + 56 \geq -35
$$

Step 2: Subtract 56 from both sides.

$$
7u \geq -35 - 56 = -91
$$

Step 3: Divide by 7.

$$
u \geq \frac{-91}{7} = -13
$$

✔ Solution: $u \geq -13$

Number Line:
Closed circle at $-13$, shade to the right.

---

6) $-2p + 4 < 7 - p$

Step 1: Move all terms with $p$ to one side and constants to the other.

Add $p$ to both sides:

$$
-2p + p + 4 < 7 \Rightarrow -p + 4 < 7
$$

Subtract 4:

$$
-p < 3
$$

Step 2: Multiply both sides by $-1$ → reverse inequality.

$$
p > -3
$$

✔ Solution: $p > -3$

Number Line:
Open circle at $-3$, shade to the right.

---

7) $4(1 - r) > 2(r + 3)$

Step 1: Expand both sides.

Left: $4 - 4r$
Right: $2r + 6$

So:
$$
4 - 4r > 2r + 6
$$

Step 2: Move all $r$ terms to one side, constants to the other.

Subtract $2r$ from both sides:
$$
4 - 6r > 6
$$

Subtract 4:
$$
-6r > 2
$$

Step 3: Divide by $-6$ → reverse inequality.

$$
r < \frac{2}{-6} = -\frac{1}{3}
$$

✔ Solution: $r < -\frac{1}{3}$

Number Line:
Open circle at $-\frac{1}{3}$, shade to the left.

---

8) $\frac{t}{4} \geq -3$

Step 1: Multiply both sides by 4.

$$
t \geq -12
$$

✔ Solution: $t \geq -12$

Number Line:
Closed circle at $-12$, shade to the right.

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✔ Summary of Solutions:

| Problem | Solution |
|--------|----------|
| 1) | $x > -2$ |
| 2) | $x \leq -\frac{7}{3}$ |
| 3) | $m < -4$ |
| 4) | $n \leq -33$ |
| 5) | $u \geq -13$ |
| 6) | $p > -3$ |
| 7) | $r < -\frac{1}{3}$ |
| 8) | $t \geq -12$ |

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📊 Number Line Representations (Descriptive):

For each solution:

- Use closed circle for $\leq$ or $\geq$
- Use open circle for $<$ or $>$
- Shade in the direction that satisfies the inequality

#### Example: For $x > -2$
```
<---o================>
-3 -2 -1 0
```

#### Example: For $x \leq -\frac{7}{3} \approx -2.33$
```
<===●----------------->
-3 -2.33 -2 0
```

You can draw similar lines for each.

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Let me know if you'd like visual diagrams or further explanation!