Question

solve the inequalities and graph the solutions.

1. (-5x + 3 < 13)
2. (2x - 1 geq 11)
3. (-3x + 3 < 24)
4. (6x - 4 leq 26)
5. (-2x - 2 geq -17)
6. (-3(4x + 1) leq -15)
7. (-x + 3 < 15)
8. (-3x + 10 > 0)

Explanation:

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1) $-5x + 3 < 13$

Step1: Subtract 3 from both sides

$-5x + 3 - 3 < 13 - 3$
$-5x < 10$

Step2: Divide by -5, flip inequality

$\frac{-5x}{-5} > \frac{10}{-5}$
$x > -2$

Graph Description:

Open circle at $-2$, arrow pointing right.
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2) $2x - 1 \geq 11$

Step1: Add 1 to both sides

$2x - 1 + 1 \geq 11 + 1$
$2x \geq 12$

Step2: Divide by 2

$\frac{2x}{2} \geq \frac{12}{2}$
$x \geq 6$

Graph Description:

Closed circle at $6$, arrow pointing right.
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3) $-3x + 3 < 24$

Step1: Subtract 3 from both sides

$-3x + 3 - 3 < 24 - 3$
$-3x < 21$

Step2: Divide by -3, flip inequality

$\frac{-3x}{-3} > \frac{21}{-3}$
$x > -7$

Graph Description:

Open circle at $-7$, arrow pointing right.
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4) $6x - 4 \leq 26$

Step1: Add 4 to both sides

$6x - 4 + 4 \leq 26 + 4$
$6x \leq 30$

Step2: Divide by 6

$\frac{6x}{6} \leq \frac{30}{6}$
$x \leq 5$

Graph Description:

Closed circle at $5$, arrow pointing left.
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5) $2x - 2 \geq -17$

Step1: Add 2 to both sides

$2x - 2 + 2 \geq -17 + 2$
$2x \geq -15$

Step2: Divide by 2

$\frac{2x}{2} \geq \frac{-15}{2}$
$x \geq -7.5$

Graph Description:

Closed circle at $-7.5$, arrow pointing right.
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6) $-3(4x + 1) \leq -15$

Step1: Divide by -3, flip inequality

$\frac{-3(4x + 1)}{-3} \geq \frac{-15}{-3}$
$4x + 1 \geq 5$

Step2: Subtract 1 from both sides

$4x + 1 - 1 \geq 5 - 1$
$4x \geq 4$

Step3: Divide by 4

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

Graph Description:

Closed circle at $1$, arrow pointing right.
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7) $-x + 3 < 15$

Step1: Subtract 3 from both sides

$-x + 3 - 3 < 15 - 3$
$-x < 12$

Step2: Multiply by -1, flip inequality

$x > -12$

Graph Description:

Open circle at $-12$, arrow pointing right.
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8) $-3x + 10 > 0$

Step1: Subtract 10 from both sides

$-3x + 10 - 10 > 0 - 10$
$-3x > -10$

Step2: Divide by -3, flip inequality

$\frac{-3x}{-3} < \frac{-10}{-3}$
$x < \frac{10}{3} \approx 3.33$

Graph Description:

Open circle at $\frac{10}{3}$, arrow pointing left.
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Answer:

1. $x > -2$
2. $x \geq 6$
3. $x > -7$
4. $x \leq 5$
5. $x \geq -7.5$
6. $x \geq 1$
7. $x > -12$
8. $x < \frac{10}{3}$

Graph Summaries:

1. Open circle at $-2$, right arrow
2. Closed circle at $6$, right arrow
3. Open circle at $-7$, right arrow
4. Closed circle at $5$, left arrow
5. Closed circle at $-7.5$, right arrow
6. Closed circle at $1$, right arrow
7. Open circle at $-12$, right arrow
8. Open circle at $\frac{10}{3}$, left arrow