Question

graph this inequality:
$x + 2y > 6$
plot points on the boundary line. select the line to switch between solid and dotted. select a region to shade it.

Explanation:

Step1: Rewrite in slope-intercept form

Rearrange the inequality to solve for $y$:

$$\begin{align*} x + 2y &> 6 \\ 2y &> -x + 6 \\ y &> -\frac{1}{2}x + 3 \end{align*}$$

Step2: Identify boundary line

The boundary is $y = -\frac{1}{2}x + 3$. Use a dotted line (since the inequality is $>$, not $\geq$). Find intercepts:

• $x$-intercept: Set $y=0$, $0 = -\frac{1}{2}x + 3 \implies x=6$, so point $(6, 0)$
• $y$-intercept: Set $x=0$, $y=3$, so point $(0, 3)$

Step3: Test a point for shading

Test $(0,0)$ in the original inequality:
$$0 + 2(0) > 6 \implies 0 > 6$$
This is false, so shade the region not containing $(0,0)$ (above the dotted line).

Answer:

1. Draw a dotted line through points $(6, 0)$ and $(0, 3)$.
2. Shade the region above this dotted line.