Question

1. $y \geq -\frac{1}{2}x + 4$
2. $y \leq \frac{5}{4}x + 2$
3. $y > 1$
4. $x \geq -3$
5. $x > 5$
6. $y < 2$

Explanation:

7) $y \geq -\frac{1}{2}x + 4$

Step1: Identify line intercepts

Find x-intercept: set $y=0$
$0 = -\frac{1}{2}x + 4 \implies x=8$
Find y-intercept: set $x=0$
$y=4$

Step2: Draw boundary line

Plot points $(8,0)$ and $(0,4)$, draw a solid line (since $\geq$ includes equality).

Step3: Shade correct region

Test $(0,0)$: $0 \geq 4$ is false. Shade the region above the line.

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8) $y \leq \frac{5}{4}x + 2$

Step1: Identify line intercepts

Find x-intercept: set $y=0$
$0 = \frac{5}{4}x + 2 \implies x=-\frac{8}{5}=-1.6$
Find y-intercept: set $x=0$
$y=2$

Step2: Draw boundary line

Plot points $(-1.6,0)$ and $(0,2)$, draw a solid line (since $\leq$ includes equality).

Step3: Shade correct region

Test $(0,0)$: $0 \leq 2$ is true. Shade the region below the line.

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9) $y > 1$

Step1: Draw boundary line

Draw a horizontal dashed line at $y=1$ (since $>$ does not include equality).

Step2: Shade correct region

Test $(0,0)$: $0 > 1$ is false. Shade the region above the line.

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10) $x \geq -3$

Step1: Draw boundary line

Draw a vertical solid line at $x=-3$ (since $\geq$ includes equality).

Step2: Shade correct region

Test $(0,0)$: $0 \geq -3$ is true. Shade the region to the right of the line.

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11) $x > 5$

Step1: Draw boundary line

Draw a vertical dashed line at $x=5$ (since $>$ does not include equality).

Step2: Shade correct region

Test $(0,0)$: $0 > 5$ is false. Shade the region to the right of the line.

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12) $y < 2$

Step1: Draw boundary line

Draw a horizontal dashed line at $y=2$ (since $<$ does not include equality).

Step2: Shade correct region

Test $(0,0)$: $0 < 2$ is true. Shade the region below the line.

Answer:

1. For $\boldsymbol{y \geq -\frac{1}{2}x + 4}$: Solid line through $(8,0)$ and $(0,4)$, shade above the line.
2. For $\boldsymbol{y \leq \frac{5}{4}x + 2}$: Solid line through $(-1.6,0)$ and $(0,2)$, shade below the line.
3. For $\boldsymbol{y > 1}$: Dashed horizontal line at $y=1$, shade above the line.
4. For $\boldsymbol{x \geq -3}$: Solid vertical line at $x=-3$, shade right of the line.
5. For $\boldsymbol{x > 5}$: Dashed vertical line at $x=5$, shade right of the line.
6. For $\boldsymbol{y < 2}$: Dashed horizontal line at $y=2$, shade below the line.