Question

7
graph the following inequality
$5x + 2y < 20$

Explanation:

Step1: Rewrite in slope-intercept form

First, isolate $y$ to match $y=mx+b$.

$$\begin{align*} 5x + 2y &< 20 \\ 2y &< -5x + 20 \\ y &< -\frac{5}{2}x + 10 \end{align*}$$

Step2: Identify boundary line

The boundary is $y=-\frac{5}{2}x + 10$. Use a dashed line since the inequality is $<$ (not inclusive).

Step3: Test a point for shading

Use the origin $(0,0)$:

$$ 0 < -\frac{5}{2}(0) + 10 \implies 0 < 10 $$

This is true, so shade the region containing $(0,0)$ (below the dashed line).

Step4: Plot key points for the line

Find intercepts:

• x-intercept: Set $y=0$:

$$ 0 = -\frac{5}{2}x + 10 \implies \frac{5}{2}x = 10 \implies x=4 $$

Point: $(4, 0)$

• y-intercept: Set $x=0$:

$$ y = -\frac{5}{2}(0) + 10 \implies y=10 $$

Point: $(0, 10)$

Answer:

1. Draw a dashed line connecting the points $(4, 0)$ and $(0, 10)$ (this is the boundary $y=-\frac{5}{2}x + 10$).
2. Shade the entire region below this dashed line (all points where $y < -\frac{5}{2}x + 10$ satisfy the inequality $5x + 2y < 20$).