Question

graph each inequality in the coordinate plane. 22. $x \leq 4$ 23. $y \geq -1$ 24. $x > -2$ 26. $-2x + y \geq 3$ 27. $x + 3y < 15$ 28. $4x - y > 2$

Explanation:

Step1: Analyze vertical/horizontal lines

For inequalities with single variables:

• 22. $x \leq 4$: Draw solid vertical line $x=4$, shade left.
• 23. $y \geq -1$: Draw solid horizontal line $y=-1$, shade above.
• 24. $x > -2$: Draw dashed vertical line $x=-2$, shade right.

Step2: Rewrite linear inequalities to slope-intercept

Rearrange to $y=mx+b$ form:

• 26. $-2x+y \geq 3 \implies y \geq 2x+3$
• 27. $x+3y < 15 \implies y < -\frac{1}{3}x+5$
• 28. $4x-y > 2 \implies y < 4x-2$

Step3: Graph linear inequalities

• 26. Solid line $y=2x+3$, shade above.
• 27. Dashed line $y=-\frac{1}{3}x+5$, shade below.
• 28. Dashed line $y=4x-2$, shade below.

Answer:

1. For $x \leq 4$: Solid vertical line at $x=4$, shade all regions where $x$ is less than or equal to 4 (left of the line).
2. For $y \geq -1$: Solid horizontal line at $y=-1$, shade all regions where $y$ is greater than or equal to -1 (above the line).
3. For $x > -2$: Dashed vertical line at $x=-2$, shade all regions where $x$ is greater than -2 (right of the line).
4. For $-2x+y \geq 3$ (or $y \geq 2x+3$): Solid line with slope 2 and y-intercept 3, shade all regions above the line.
5. For $x+3y < 15$ (or $y < -\frac{1}{3}x+5$): Dashed line with slope $-\frac{1}{3}$ and y-intercept 5, shade all regions below the line.
6. For $4x-y > 2$ (or $y < 4x-2$): Dashed line with slope 4 and y-intercept -2, shade all regions below the line.