Question

which of the following equations match the given graph?
$y > -\frac{3}{4}x + 3$
$y \leq -\frac{3}{4}x + 3$
$y < -\frac{3}{4}x + 3$
$y \geq -\frac{3}{4}x + 3$

Explanation:

Step1: Identify line equation

The line has slope $-\frac{3}{4}$ and y-intercept 3, so its equation is $y = -\frac{3}{4}x + 3$.

Step2: Check line style

The line is solid, so the inequality uses $\leq$ or $\geq$.

Step3: Test shaded region

Pick a point in the shaded area, e.g., $(0,3)$:
Substitute into $y \geq -\frac{3}{4}x + 3$:
$3 \geq -\frac{3}{4}(0) + 3$
$3 \geq 3$, which is true.
Substitute into $y \leq -\frac{3}{4}x + 3$:
$3 \leq 3$ is true, but test another point like $(4,0)$:
For $y \geq -\frac{3}{4}x + 3$: $0 \geq -\frac{3}{4}(4)+3 \to 0 \geq 0$, true.
For $y \leq -\frac{3}{4}x + 3$: $0 \leq 0$, true. Test $(0,4)$ (above line, shaded):
$4 \geq -\frac{3}{4}(0)+3 \to 4 \geq 3$, true.
$4 \leq 3$ is false, so the inequality is $y \geq -\frac{3}{4}x + 3$.

Answer:

$y \geq -\frac{3}{4}x + 3$