Question

practice
solve each inequality. graph the solution set on a number line (examples 1-3)

1. $-3x - 3 > 12$
2. $-4 \leq 4x + 8$
3. $6.5x - 11.3 \leq 8.2$
4. $-2.45x + 3.2 < -6.6$
5. $\frac{5}{8} > \frac{1}{2}x - \frac{1}{4}$
6. $\frac{1}{5} \leq \frac{x}{10} + \frac{1}{4}$
7. $5(x - 3) > 15$
8. $-8(x + 7.5) \leq 10$

Explanation:

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Problem 1: $-3x - 3 > 12$

Step1: Add 3 to both sides

$-3x - 3 + 3 > 12 + 3$
$-3x > 15$

Step2: Divide by -3, reverse inequality

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

Graph:

Draw an open circle at $x=-5$ on the number line, shade all values to the left.

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Problem 2: $-4 \leq 4x + 8$

Step1: Subtract 8 from both sides

$-4 - 8 \leq 4x + 8 - 8$
$-12 \leq 4x$

Step2: Divide by 4

$\frac{-12}{4} \leq \frac{4x}{4}$
$-3 \leq x$ or $x \geq -3$

Graph:

Draw a closed circle at $x=-3$ on the number line, shade all values to the right.

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Problem 3: $6.5x - 11.3 \leq 8.2$

Step1: Add 11.3 to both sides

$6.5x - 11.3 + 11.3 \leq 8.2 + 11.3$
$6.5x \leq 19.5$

Step2: Divide by 6.5

$\frac{6.5x}{6.5} \leq \frac{19.5}{6.5}$
$x \leq 3$

Graph:

Draw a closed circle at $x=3$ on the number line, shade all values to the left.

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Problem 4: $-2.45x + 3.2 < -6.6$

Step1: Subtract 3.2 from both sides

$-2.45x + 3.2 - 3.2 < -6.6 - 3.2$
$-2.45x < -9.8$

Step2: Divide by -2.45, reverse inequality

$\frac{-2.45x}{-2.45} > \frac{-9.8}{-2.45}$
$x > 4$

Graph:

Draw an open circle at $x=4$ on the number line, shade all values to the right.

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Problem 5: $\frac{5}{8} > \frac{1}{2}x - \frac{1}{4}$

Step1: Add $\frac{1}{4}$ to both sides

$\frac{5}{8} + \frac{1}{4} > \frac{1}{2}x$
$\frac{5}{8} + \frac{2}{8} > \frac{1}{2}x$
$\frac{7}{8} > \frac{1}{2}x$

Step2: Multiply by 2

$\frac{7}{8} \times 2 > x$
$x < \frac{7}{4}$ or $x < 1.75$

Graph:

Draw an open circle at $x=1.75$ on the number line, shade all values to the left.

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Problem 6: $\frac{1}{5} \leq \frac{x}{10} + \frac{1}{4}$

Step1: Subtract $\frac{1}{4}$ from both sides

$\frac{1}{5} - \frac{1}{4} \leq \frac{x}{10}$
$\frac{4}{20} - \frac{5}{20} \leq \frac{x}{10}$
$-\frac{1}{20} \leq \frac{x}{10}$

Step2: Multiply by 10

$-\frac{1}{20} \times 10 \leq x$
$x \leq -\frac{1}{2}$ or $x \geq -0.5$

Graph:

Draw a closed circle at $x=-0.5$ on the number line, shade all values to the right.

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Problem 7: $5(x - 3) > 15$

Step1: Divide both sides by 5

$\frac{5(x - 3)}{5} > \frac{15}{5}$
$x - 3 > 3$

Step2: Add 3 to both sides

$x - 3 + 3 > 3 + 3$
$x > 6$

Graph:

Draw an open circle at $x=6$ on the number line, shade all values to the right.

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Problem 8: $-8(x + 7.5) \leq 10$

Step1: Divide by -8, reverse inequality

$\frac{-8(x + 7.5)}{-8} \geq \frac{10}{-8}$
$x + 7.5 \geq -1.25$

Step2: Subtract 7.5 from both sides

$x + 7.5 - 7.5 \geq -1.25 - 7.5$
$x \geq -8.75$

Graph:

Draw a closed circle at $x=-8.75$ on the number line, shade all values to the right.

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Answer:

1. $x < -5$
2. $x \geq -3$
3. $x \leq 3$
4. $x > 4$
5. $x < \frac{7}{4}$
6. $x \geq -\frac{1}{2}$
7. $x > 6$
8. $x \geq -8.75$

(Graphs follow the instructions provided for each inequality above)