Question

1. graph the system of inequalities: $y \geq 4$ $y < 2x + 6$

Explanation:

Step1: Graph boundary $y=4$

Draw a solid horizontal line at $y=4$ (solid because the inequality is $\geq$, so the line is included). Shade the region above this line, since $y$ is greater than or equal to 4.

Step2: Graph boundary $y=2x+6$

Draw a dashed line for $y=2x+6$ (dashed because the inequality is $<$, so the line is not included). This line has a y-intercept at $(0,6)$ and a slope of $2$.

Step3: Shade correct region for $y<2x+6$

Choose a test point not on the line, e.g., $(0,0)$. Substitute into the inequality: $0 < 2(0)+6$ → $0<6$, which is true. So shade the region below the dashed line.

Step4: Identify overlapping shaded region

The solution to the system is the area that is shaded in both steps 1 and 3: the region above the solid line $y=4$ and below the dashed line $y=2x+6$.

Answer:

The graph consists of:

1. A solid horizontal line $y=4$ with shading above it.
2. A dashed line $y=2x+6$ (y-intercept $(0,6)$, slope $2$) with shading below it.
3. The overlapping shaded region (above $y=4$ and below $y=2x+6$) is the solution set.