Question

solve the inequalities and graph the solutions.

1. \\( 4x + 9 < 13 \\)
2. \\( -3x - 5 \leq 7 \\)
3. \\( -2x + 3 \geq 23 \\)
4. \\( 8x - 5 > 27 \\)

challenge: solve the inequality and graph the solution.

1. \\( -6 < 2x < 10 \\)

Explanation:

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1) Inequality: $4x + 9 < 13$

Step1: Subtract 9 from both sides

$4x + 9 - 9 < 13 - 9$
$4x < 4$

Step2: Divide by 4 on both sides

$\frac{4x}{4} < \frac{4}{4}$
$x < 1$

Graph Description:

Draw an open circle at $1$ on the number line, then shade all values to the left of $1$.

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2) Inequality: $-3x - 5 \leq 7$

Step1: Add 5 to both sides

$-3x - 5 + 5 \leq 7 + 5$
$-3x \leq 12$

Step2: Divide by -3 (reverse inequality)

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

Graph Description:

Draw a closed circle at $-4$ on the number line, then shade all values to the right of $-4$.

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3) Inequality: $-2x + 3 \geq 23$

Step1: Subtract 3 from both sides

$-2x + 3 - 3 \geq 23 - 3$
$-2x \geq 20$

Step2: Divide by -2 (reverse inequality)

$\frac{-2x}{-2} \leq \frac{20}{-2}$
$x \leq -10$

Graph Description:

Draw a closed circle at $-10$ on the number line, then shade all values to the left of $-10$.

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4) Inequality: $8x - 5 > 27$

Step1: Add 5 to both sides

$8x - 5 + 5 > 27 + 5$
$8x > 32$

Step2: Divide by 8 on both sides

$\frac{8x}{8} > \frac{32}{8}$
$x > 4$

Graph Description:

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

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5) Inequality: $-6 < 2x < 10$

Step1: Divide all parts by 2

$\frac{-6}{2} < \frac{2x}{2} < \frac{10}{2}$
$-3 < x < 5$

Graph Description:

Draw open circles at $-3$ and $5$ on the number line, then shade all values between $-3$ and $5$.

Answer:

1. $x < 1$
2. $x \geq -4$
3. $x \leq -10$
4. $x > 4$
5. $-3 < x < 5$

(Graphs follow the descriptions provided above for each inequality)