Solving Inequalities - Kuta Software - Free Printable

Let's solve each inequality step by step and graph the solution on the number line. I'll go through each problem one at a time.

---

1) $ 0 > 3x - 6 $

Step 1: Add 6 to both sides:
$$
0 + 6 > 3x \Rightarrow 6 > 3x
$$

Step 2: Divide both sides by 3:
$$
2 > x \quad \text{or} \quad x < 2
$$

✔ Solution: $ x < 2 $

Graph: Open circle at 2, shade to the left.

---

2) $ 4x + 1 - 1 \geq -8 $

Simplify:
$$
4x + 0 \geq -8 \Rightarrow 4x \geq -8
$$

Divide by 4:
$$
x \geq -2
$$

✔ Solution: $ x \geq -2 $

Graph: Closed circle at -2, shade to the right.

---

3) $ -1 \leq 2n + 4 - 5 $

Simplify right side:
$$
-1 \leq 2n - 1
$$

Add 1 to both sides:
$$
0 \leq 2n \Rightarrow 2n \geq 0
$$

Divide by 2:
$$
n \geq 0
$$

✔ Solution: $ n \geq 0 $

Graph: Closed circle at 0, shade to the right.

---

4) $ -6 > 5n + 5 + 4 $

Simplify:
$$
-6 > 5n + 9
$$

Subtract 9 from both sides:
$$
-15 > 5n
$$

Divide by 5:
$$
-3 > n \quad \text{or} \quad n < -3
$$

✔ Solution: $ n < -3 $

Graph: Open circle at -3, shade to the left.

---

5) $ 0 \leq 2n + 3n $

Combine like terms:
$$
0 \leq 5n
$$

Divide by 5:
$$
0 \leq n \quad \text{or} \quad n \geq 0
$$

✔ Solution: $ n \geq 0 $

Graph: Closed circle at 0, shade to the right.

---

6) $ 2p - 4p \leq -2 $

Combine like terms:
$$
-2p \leq -2
$$

Divide by -2 → reverse inequality sign:
$$
p \geq 1
$$

✔ Solution: $ p \geq 1 $

Graph: Closed circle at 1, shade to the right.

---

7) $ 7 < -(k - 3) + 2 $

Distribute the negative:
$$
7 < -k + 3 + 2 \Rightarrow 7 < -k + 5
$$

Subtract 5 from both sides:
$$
2 < -k
$$

Multiply both sides by -1 → reverse inequality:
$$
-2 > k \quad \text{or} \quad k < -2
$$

✔ Solution: $ k < -2 $

Graph: Open circle at -2, shade to the left.

---

8) $ 3 - 2(n - 4) > -1 $

Distribute:
$$
3 - 2n + 8 > -1 \Rightarrow 11 - 2n > -1
$$

Subtract 11:
$$
-2n > -12
$$

Divide by -2 → reverse inequality:
$$
n < 6
$$

✔ Solution: $ n < 6 $

Graph: Open circle at 6, shade to the left.

---

9) $ -5(1 - 4a) > -5 $

Distribute:
$$
-5 + 20a > -5
$$

Add 5 to both sides:
$$
20a > 0
$$

Divide by 20:
$$
a > 0
$$

✔ Solution: $ a > 0 $

Graph: Open circle at 0, shade to the right.

---

10) $ -2(b + 1) + 4 < 10 $

Distribute:
$$
-2b - 2 + 4 < 10 \Rightarrow -2b + 2 < 10
$$

Subtract 2:
$$
-2b < 8
$$

Divide by -2 → reverse inequality:
$$
b > -4
$$

✔ Solution: $ b > -4 $

Graph: Open circle at -4, shade to the right.

---

✔ Final Answers Summary:

| Problem | Solution | Graph Description |
|--------|---------|-------------------|
| 1 | $ x < 2 $ | Open circle at 2, left |
| 2 | $ x \geq -2 $ | Closed circle at -2, right |
| 3 | $ n \geq 0 $ | Closed circle at 0, right |
| 4 | $ n < -3 $ | Open circle at -3, left |
| 5 | $ n \geq 0 $ | Closed circle at 0, right |
| 6 | $ p \geq 1 $ | Closed circle at 1, right |
| 7 | $ k < -2 $ | Open circle at -2, left |
| 8 | $ n < 6 $ | Open circle at 6, left |
| 9 | $ a > 0 $ | Open circle at 0, right |
|10 | $ b > -4 $ | Open circle at -4, right |

> 📌 Remember:
> - Use open circles for $ < $ or $ > $
> - Use closed circles for $ \leq $ or $ \geq $
> - Always reverse the inequality when multiplying/dividing by a negative number

Let me know if you'd like a visual representation of any graph!