Question

choose all that correctly graph the given inequality.

Explanation:

Step1: Analyze Inequality A

Inequality: $y \geq -5x - 3$

1. Identify the boundary line: $y=-5x-3$. The y-intercept is $(0,-3)$, x-intercept is when $y=0$: $0=-5x-3 \implies x=-\frac{3}{5}=-0.6$. The line has a negative slope, matches the graph in A.
2. Test a point (e.g., $(0,0)$): $0 \geq -5(0)-3 \implies 0 \geq -3$, which is true. The region containing $(0,0)$ is above the line, which matches the shaded area in A. So A is correct.

Step2: Analyze Inequality B

Inequality: $y \leq 3x + 2$

1. Identify the boundary line: $y=3x+2$. The y-intercept is $(0,2)$, x-intercept is when $y=0$: $0=3x+2 \implies x=-\frac{2}{3}\approx-0.67$. The line has a positive slope, but the graph in B has a negative slope line. So B is incorrect.

Step3: Analyze Inequality C

Inequality: $y \geq \frac{1}{2}x + 1$

1. Identify the boundary line: $y=\frac{1}{2}x+1$. The y-intercept is $(0,1)$, x-intercept is when $y=0$: $0=\frac{1}{2}x+1 \implies x=-2$. The line has a positive slope, but the graph in C has a negative slope line. So C is incorrect.

Step4: Analyze Inequality D

Inequality: $y \leq \frac{1}{5}x + 2$

1. Identify the boundary line: $y=\frac{1}{5}x+2$. The y-intercept is $(0,2)$, x-intercept is when $y=0$: $0=\frac{1}{5}x+2 \implies x=-10$. The line has a small positive slope, but the graph in D has a steep negative slope line. So D is incorrect.

Answer:

A. $y \geq -5x - 3$ (correct graph)