Question

solving & graphing inequalities
solve each inequality and graph the solution.

1. $3x - 10 < 2$
2. $5x + 9 \geq 24$
3. $\frac{x - 4}{2} > 3$
4. $8x - 9 \leq 15$
5. $9x - 1 > 17$
6. $\frac{x - 3}{9} \geq 7$
7. $\frac{x}{6} + 2 \leq 5$
8. $4 + 2x < 18$

Explanation:

1) Step1: Add 10 to both sides

$3x - 10 + 10 < 2 + 10$
$3x < 12$

1) Step2: Divide by 3 on both sides

$\frac{3x}{3} < \frac{12}{3}$
$x < 4$
---

2) Step1: Subtract 9 from both sides

$5x + 9 - 9 \geq 24 - 9$
$5x \geq 15$

2) Step2: Divide by 5 on both sides

$\frac{5x}{5} \geq \frac{15}{5}$
$x \geq 3$
---

3) Step1: Multiply by 2 on both sides

$2 \times \frac{x-4}{2} > 3 \times 2$
$x - 4 > 6$

3) Step2: Add 4 to both sides

$x - 4 + 4 > 6 + 4$
$x > 10$
---

4) Step1: Add 9 to both sides

$8x - 9 + 9 \leq 15 + 9$
$8x \leq 24$

4) Step2: Divide by 8 on both sides

$\frac{8x}{8} \leq \frac{24}{8}$
$x \leq 3$
---

5) Step1: Add 1 to both sides

$9x - 1 + 1 > 17 + 1$
$9x > 18$

5) Step2: Divide by 9 on both sides

$\frac{9x}{9} > \frac{18}{9}$
$x > 2$
---

6) Step1: Multiply by 9 on both sides

$9 \times \frac{x-3}{9} \geq 7 \times 9$
$x - 3 \geq 63$

6) Step2: Add 3 to both sides

$x - 3 + 3 \geq 63 + 3$
$x \geq 66$
---

7) Step1: Subtract 2 from both sides

$\frac{x}{6} + 2 - 2 \leq 5 - 2$
$\frac{x}{6} \leq 3$

7) Step2: Multiply by 6 on both sides

$6 \times \frac{x}{6} \leq 3 \times 6$
$x \leq 18$
---

8) Step1: Subtract 4 from both sides

$4 + 2x - 4 < 18 - 4$
$2x < 14$

8) Step2: Divide by 2 on both sides

$\frac{2x}{2} < \frac{14}{2}$
$x < 7$

Answer:

1. $x < 4$ (Graph: Open circle at 4, arrow left)
2. $x \geq 3$ (Graph: Closed circle at 3, arrow right)
3. $x > 10$ (Graph: Open circle at 10, arrow right)
4. $x \leq 3$ (Graph: Closed circle at 3, arrow left)
5. $x > 2$ (Graph: Open circle at 2, arrow right)
6. $x \geq 66$ (Graph: Closed circle at 66, arrow right)
7. $x \leq 18$ (Graph: Closed circle at 18, arrow left)
8. $x < 7$ (Graph: Open circle at 7, arrow left)