Question

graph each linear inequality.

1. ( y leq x - 1 )
2. ( y geq 3x - 2 )
3. ( y < -4x - 1 )
4. ( y > 2x - 6 )
5. ( y < 5x - 5 )
6. ( y geq \frac{1}{2}x - 3 )
7. ( y > -3x )
8. ( y geq -x )

see problem 2.

Explanation:

Step1: Analyze boundary line type

For each inequality:

• $\leq$ / $\geq$: Solid boundary line (equal included)
• $<$ / $>$: Dashed boundary line (equal excluded)

Step2: Graph boundary line

For each $y = mx + b$:

1. Plot y-intercept $(0, b)$
2. Use slope $m=\frac{\text{rise}}{\text{run}}$ to plot 2nd point
3. Draw line (solid/dashed per Step1)

Step3: Test point for shading

Use $(0,0)$ (if not on line):

• Substitute into inequality. If true, shade side with $(0,0)$; if false, shade opposite side.

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For 14. $y \leq x - 1$

Step1: Solid boundary line

$y = x - 1$ (solid, $\leq$)

Step2: Plot boundary line

Y-intercept: $(0, -1)$, slope $m=1$. Plot $(1,0)$, draw solid line.

Step3: Shade correct region

Test $(0,0)$: $0 \leq 0 -1 \to 0 \leq -1$ (false). Shade below line.

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For 15. $y \geq 3x - 2$

Step1: Solid boundary line

$y = 3x - 2$ (solid, $\geq$)

Step2: Plot boundary line

Y-intercept: $(0, -2)$, slope $m=3$. Plot $(1,1)$, draw solid line.

Step3: Shade correct region

Test $(0,0)$: $0 \geq 0 -2 \to 0 \geq -2$ (true). Shade above line.

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For 16. $y < -4x - 1$

Step1: Dashed boundary line

$y = -4x - 1$ (dashed, $<$)

Step2: Plot boundary line

Y-intercept: $(0, -1)$, slope $m=-4$. Plot $(1, -5)$, draw dashed line.

Step3: Shade correct region

Test $(0,0)$: $0 < 0 -1 \to 0 < -1$ (false). Shade below line.

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For 17. $y > 2x - 6$

Step1: Dashed boundary line

$y = 2x - 6$ (dashed, $>$)

Step2: Plot boundary line

Y-intercept: $(0, -6)$, slope $m=2$. Plot $(3,0)$, draw dashed line.

Step3: Shade correct region

Test $(0,0)$: $0 > 0 -6 \to 0 > -6$ (true). Shade above line.

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For 18. $y < 5x - 5$

Step1: Dashed boundary line

$y = 5x - 5$ (dashed, $<$)

Step2: Plot boundary line

Y-intercept: $(0, -5)$, slope $m=5$. Plot $(1,0)$, draw dashed line.

Step3: Shade correct region

Test $(0,0)$: $0 < 0 -5 \to 0 < -5$ (false). Shade below line.

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For 19. $y \leq \frac{1}{2}x - 3$

Step1: Solid boundary line

$y = \frac{1}{2}x - 3$ (solid, $\leq$)

Step2: Plot boundary line

Y-intercept: $(0, -3)$, slope $m=\frac{1}{2}$. Plot $(2, -2)$, draw solid line.

Step3: Shade correct region

Test $(0,0)$: $0 \leq 0 -3 \to 0 \leq -3$ (false). Shade below line.

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For 20. $y > -3x$

Step1: Dashed boundary line

$y = -3x$ (dashed, $>$)

Step2: Plot boundary line

Y-intercept: $(0,0)$, slope $m=-3$. Plot $(1, -3)$, draw dashed line.

Step3: Shade correct region

Test $(1,0)$: $0 > -3(1) \to 0 > -3$ (true). Shade above line.

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For 21. $y \geq -x$

Step1: Solid boundary line

$y = -x$ (solid, $\geq$)

Step2: Plot boundary line

Y-intercept: $(0,0)$, slope $m=-1$. Plot $(1, -1)$, draw solid line.

Step3: Shade correct region

Test $(0,1)$: $1 \geq 0$ (true). Shade above line.

Answer:

1. Solid line $y=x-1$, shade below the line.
2. Solid line $y=3x-2$, shade above the line.
3. Dashed line $y=-4x-1$, shade below the line.
4. Dashed line $y=2x-6$, shade above the line.
5. Dashed line $y=5x-5$, shade below the line.
6. Solid line $y=\frac{1}{2}x-3$, shade below the line.
7. Dashed line $y=-3x$, shade above the line.
8. Solid line $y=-x$, shade above the line.