Question

solve the following system of inequalities graphically on the set of axes below. state the coordinates of a point in the solution set. $y > -2x - 1$ $y < \frac{1}{2}x + 4$ (graph of coordinate plane with x from -10 to 10 and y from -10 to 10)

Explanation:

Step1: Graph first inequality boundary

First, identify the boundary line $y=-2x-1$. This is a line with slope $-2$ and y-intercept $(0,-1)$. Since the inequality is $y > -2x -1$, draw a dashed line (because the inequality does not include equality) and shade the region above the line.

Step2: Graph second inequality boundary

Next, identify the boundary line $y=\frac{1}{2}x+4$. This is a line with slope $\frac{1}{2}$ and y-intercept $(0,4)$. Since the inequality is $y < \frac{1}{2}x +4$, draw a dashed line and shade the region below the line.

Step3: Identify overlapping shaded region

The solution set is the area where the two shaded regions overlap. To find a point in this set, pick any point that satisfies both inequalities. Test $(0,0)$:
For $y > -2x -1$: $0 > -2(0)-1 \implies 0 > -1$, which is true.
For $y < \frac{1}{2}x +4$: $0 < \frac{1}{2}(0)+4 \implies 0 < 4$, which is true.

Answer:

A point in the solution set is $(0,0)$ (any point in the overlapping shaded region is valid).