Question

5.7 practice
sketch the graph of each linear inequality.

1. $x+y<4$
2. $3x+y>3$
3. $5x-2y\leq-10$
4. $x-2y\leq-2$
5. $y
6. $y\leq-2x+1$

Explanation:

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1) $x+y<4$

Step1: Rewrite to slope-intercept form

$y < -x + 4$

Step2: Identify boundary line

Boundary: $y=-x+4$ (dashed, since $<$)

Step3: Test origin $(0,0)$

$0+0<4$ → $0<4$, true. Shade region containing $(0,0)$.

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2) $3x+y>3$

Step1: Rewrite to slope-intercept form

$y > -3x + 3$

Step2: Identify boundary line

Boundary: $y=-3x+3$ (dashed, since $>$)

Step3: Test origin $(0,0)$

$0+0>3$ → $0>3$, false. Shade region opposite of $(0,0)$.

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3) $5x-2y\leq-10$

Step1: Rewrite to slope-intercept form

$-2y \leq -5x -10$ → $y \geq \frac{5}{2}x + 5$

Step2: Identify boundary line

Boundary: $y=\frac{5}{2}x+5$ (solid, since $\leq$)

Step3: Test origin $(0,0)$

$0-0\leq-10$ → $0\leq-10$, false. Shade region opposite of $(0,0)$.

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4) $x-2y\leq-2$

Step1: Rewrite to slope-intercept form

$-2y \leq -x -2$ → $y \geq \frac{1}{2}x + 1$

Step2: Identify boundary line

Boundary: $y=\frac{1}{2}x+1$ (solid, since $\leq$)

Step3: Test origin $(0,0)$

$0-0\leq-2$ → $0\leq-2$, false. Shade region opposite of $(0,0)$.

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5) $y

Step1: Identify boundary line

Boundary: $y=x-2$ (dashed, since $<$)

Step2: Test origin $(0,0)$

$0<0-2$ → $0<-2$, false. Shade region opposite of $(0,0)$.

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6) $y\leq-2x+1$

Step1: Identify boundary line

Boundary: $y=-2x+1$ (solid, since $\leq$)

Step2: Test origin $(0,0)$

$0\leq0+1$ → $0\leq1$, true. Shade region containing $(0,0)$.

Answer:

1. Graph for $x+y<4$: Dashed line $y=-x+4$, shade below/left of the line.
2. Graph for $3x+y>3$: Dashed line $y=-3x+3$, shade above/left of the line.
3. Graph for $5x-2y\leq-10$: Solid line $y=\frac{5}{2}x+5$, shade above the line.
4. Graph for $x-2y\leq-2$: Solid line $y=\frac{1}{2}x+1$, shade above the line.
5. Graph for $y: Dashed line $y=x-2$, shade below the line.
6. Graph for $y\leq-2x+1$: Solid line $y=-2x+1$, shade below the line.

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Key Graphing Notes:

• Dashed lines are used for $<$ or $>$ (boundary not included)
• Solid lines are used for $\leq$ or $\geq$ (boundary included)
• Shading direction is determined by testing a point (e.g., the origin) in the original inequality.