Question

directions: show all work in order to receive full

1. graph:

$y \leq \frac{1}{2}x + 2$
$y < - 2x - 3$
(graph grid with x and y axes)

Explanation:

Step1: Analyze first inequality line

The inequality $y \leq \frac{1}{2}x + 2$ uses a solid line (due to $\leq$) from the equation $y=\frac{1}{2}x + 2$.

• y-intercept: $(0, 2)$
• Slope: $\frac{1}{2}$ (rise 1, run 2)

Shade below the line (since $y \leq$).

Step2: Analyze second inequality line

The inequality $y < -2x - 3$ uses a dashed line (due to $<$) from the equation $y=-2x - 3$.

• y-intercept: $(0, -3)$
• Slope: $-2$ (rise -2, run 1)

Shade below the line (since $y <$).

Step3: Find intersection point

Set $\frac{1}{2}x + 2 = -2x - 3$ to find where lines cross:
$$\frac{1}{2}x + 2x = -3 - 2$$
$$\frac{5}{2}x = -5$$
$$x = -2$$
Substitute $x=-2$ into $y=-2x - 3$:
$$y = -2(-2) - 3 = 4 - 3 = 1$$
Intersection: $(-2, 1)$

Step4: Identify solution region

The solution is the overlapping shaded area: below the solid line $y=\frac{1}{2}x + 2$ AND below the dashed line $y=-2x - 3$.

Answer:

1. Draw a solid line for $y=\frac{1}{2}x + 2$ (passing through $(0,2)$ and $(2,3)$), shade below it.
2. Draw a dashed line for $y=-2x - 3$ (passing through $(0,-3)$ and $(1,-5)$), shade below it.
3. The overlapping shaded area (including points on the solid line where they overlap with the shaded region) is the solution set, bounded by the two lines intersecting at $(-2, 1)$.