Question

1. $y \geq \frac{1}{4}x - 5$

Explanation:

Step1: Identify boundary line

The inequality is $y \geq \frac{1}{4}x - 5$, so the boundary line is $y = \frac{1}{4}x - 5$. Since the inequality is $\geq$, the line will be solid.

Step2: Find intercepts of boundary

• x-intercept: Set $y=0$, solve for $x$:

$0 = \frac{1}{4}x - 5$
$\frac{1}{4}x = 5$
$x = 20$
So the x-intercept is $(20, 0)$.

• y-intercept: Set $x=0$, solve for $y$:

$y = \frac{1}{4}(0) - 5 = -5$
So the y-intercept is $(0, -5)$.

Step3: Plot boundary line

Draw a solid line connecting $(20, 0)$ and $(0, -5)$, extending across the grid.

Step4: Test a point for shading

Choose a test point not on the line, e.g., $(0,0)$:
Substitute into $y \geq \frac{1}{4}x - 5$:
$0 \geq \frac{1}{4}(0) - 5$
$0 \geq -5$, which is true. So shade the region that includes $(0,0)$ (above the boundary line).

Answer:

1. Draw a solid line for $y = \frac{1}{4}x - 5$ using intercepts $(20, 0)$ and $(0, -5)$.
2. Shade the entire region above (and including) this solid line.