Question

trazar el gráfico de la desigualdad.
$x + 2y \geq 8$

Explanation:

Step1: Rewrite to slope-intercept form

First, isolate $y$ to identify the line:

$$\begin{align*} x + 2y &\geq 8 \\ 2y &\geq -x + 8 \\ y &\geq -\frac{1}{2}x + 4 \end{align*}$$

Step2: Graph the boundary line

The boundary is $y = -\frac{1}{2}x + 4$.

• Y-intercept: $(0, 4)$
• X-intercept (set $y=0$):

$$ 0 = -\frac{1}{2}x + 4 \implies x = 8 $$

So the line passes through $(0, 4)$ and $(8, 0)$. Draw a solid line (since the inequality is $\geq$, the boundary is included).

Step3: Shade the correct region

Test the origin $(0,0)$ in the inequality:

$$ 0 + 2(0) \geq 8 \implies 0 \geq 8 $$

This is false, so shade the region above and to the right of the solid line (the region that does not contain the origin).

Answer:

1. Draw a solid straight line connecting points $(0, 4)$ and $(8, 0)$.
2. Shade the area above and to the right of this solid line.