Question

1. $12x + 8y \leq 24$

$8y < -12x + 24$
$\frac{8y}{8} < \frac{-12x}{8} + \frac{24}{8}$
$y < -\frac{12}{8}x + 3$

1. $y \leq 1$

Explanation:

Step1: Simplify the inequality

Rearrange to slope-intercept form:
$$12x + 8y \leq 24$$
$$8y \leq -12x + 24$$
$$y \leq -\frac{12}{8}x + 3$$
$$y \leq -\frac{3}{2}x + 3$$

Step2: Find intercepts for the line

Calculate x-intercept (set $y=0$):
$$0 = -\frac{3}{2}x + 3$$
$$\frac{3}{2}x = 3$$
$$x = 2$$
So x-intercept is $(2, 0)$

Calculate y-intercept (set $x=0$):
$$y = -\frac{3}{2}(0) + 3$$
$$y = 3$$
So y-intercept is $(0, 3)$

Step3: Determine line style and shading

Since the inequality is $\leq$, use a solid line for $y = -\frac{3}{2}x + 3$. Shade the region below the line (as $y$ is less than or equal to the line).

Answer:

1. Draw a solid line connecting the points $(2, 0)$ and $(0, 3)$.
2. Shade the entire region that lies below this solid line (including the line itself).