Question

graph each inequality in the coordinate plane.

1. $x \leq 4$
2. $y \geq -1$
3. $x > -2$
4. $y < -4$
5. $-2x + y \geq 3$
6. $x + 3y < 15$
7. $4x - y > 2$
8. $-x + 0.25y \leq -1.75$

see problems 3 and 4.

Explanation:

22. $x \leq 4$

Step1: Identify boundary line

Boundary is $x=4$, a vertical line.

Step2: Determine line style

$\leq$ means solid line.

Step3: Shade correct region

Shade left of $x=4$ (where $x$ is smaller).

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23. $y \geq -1$

Step1: Identify boundary line

Boundary is $y=-1$, a horizontal line.

Step2: Determine line style

$\geq$ means solid line.

Step3: Shade correct region

Shade above $y=-1$ (where $y$ is larger).

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24. $x > -2$

Step1: Identify boundary line

Boundary is $x=-2$, a vertical line.

Step2: Determine line style

$>$ means dashed line.

Step3: Shade correct region

Shade right of $x=-2$ (where $x$ is larger).

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25. $y < -4$

Step1: Identify boundary line

Boundary is $y=-4$, a horizontal line.

Step2: Determine line style

$<$ means dashed line.

Step3: Shade correct region

Shade below $y=-4$ (where $y$ is smaller).

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26. $-2x + y \geq 3$

Step1: Rewrite in slope-intercept form

$y \geq 2x + 3$

Step2: Identify boundary line

Boundary: $y=2x+3$, slope $2$, y-intercept $(0,3)$.

Step3: Determine line style

$\geq$ means solid line.

Step4: Shade correct region

Shade above the line (test $(0,0)$: $0 \geq 3$ is false, so opposite side).

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27. $x + 3y < 15$

Step1: Rewrite in slope-intercept form

$y < -\frac{1}{3}x + 5$

Step2: Identify boundary line

Boundary: $y=-\frac{1}{3}x+5$, slope $-\frac{1}{3}$, y-intercept $(0,5)$.

Step3: Determine line style

$<$ means dashed line.

Step4: Shade correct region

Shade below the line (test $(0,0)$: $0 < 15$ is true, so this side).

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28. $4x - y > 2$

Step1: Rewrite in slope-intercept form

$y < 4x - 2$

Step2: Identify boundary line

Boundary: $y=4x-2$, slope $4$, y-intercept $(0,-2)$.

Step3: Determine line style

$<$ (rewritten) means dashed line.

Step4: Shade correct region

Shade below the line (test $(0,0)$: $0 > 2$ is false, so opposite side).

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29. $-x + 0.25y \leq -1.75$

Step1: Rewrite in slope-intercept form

Multiply by 4: $-4x + y \leq -7 \implies y \leq 4x - 7$

Step2: Identify boundary line

Boundary: $y=4x-7$, slope $4$, y-intercept $(0,-7)$.

Step3: Determine line style

$\leq$ means solid line.

Step4: Shade correct region

Shade below the line (test $(0,0)$: $0 \leq -7$ is false, so opposite side).

Answer:

1. For $\boldsymbol{x \leq 4}$: Draw a solid vertical line at $x=4$, shade all areas to the left of the line.
2. For $\boldsymbol{y \geq -1}$: Draw a solid horizontal line at $y=-1$, shade all areas above the line.
3. For $\boldsymbol{x > -2}$: Draw a dashed vertical line at $x=-2$, shade all areas to the right of the line.
4. For $\boldsymbol{y < -4}$: Draw a dashed horizontal line at $y=-4$, shade all areas below the line.
5. For $\boldsymbol{-2x + y \geq 3}$: Draw a solid line for $y=2x+3$, shade all areas above this line.
6. For $\boldsymbol{x + 3y < 15}$: Draw a dashed line for $y=-\frac{1}{3}x+5$, shade all areas below this line.
7. For $\boldsymbol{4x - y > 2}$: Draw a dashed line for $y=4x-2$, shade all areas below this line.
8. For $\boldsymbol{-x + 0.25y \leq -1.75}$: Draw a solid line for $y=4x-7$, shade all areas below this line.