Question

lesson 3: rsg
a line connected to each linear inequality is graphed to divide the plane into two half planes. test points on each side of the line to determine which half plane contains all of the solutions and then shade the half plane containing the solutions.

1. $y \leq 3x + 4$
2. $y < 7x - 2$
3. $y > \frac{-3}{5}x + 2$
4. $y\geq - 6$

Explanation:

Step1: Analyze Q13 boundary & test point

Inequality: $y \leq 3x + 4$. Boundary line $y=3x+4$ is solid (due to $\leq$). Test $(0,0)$: $0 \leq 3(0)+4 \to 0 \leq 4$, true.

Step2: Shade Q13 valid half-plane

Shade the half-plane containing $(0,0)$ (below/on the solid line).

Step3: Analyze Q14 boundary & test point

Inequality: $y < 7x - 2$. Boundary line $y=7x-2$ is dashed (due to $<$). Test $(0,0)$: $0 < 7(0)-2 \to 0 < -2$, false.

Step4: Shade Q14 valid half-plane

Shade the half-plane opposite $(0,0)$ (below the dashed line).

Step5: Analyze Q15 boundary & test point

Inequality: $y > \frac{-3}{5}x + 2$. Boundary line $y=\frac{-3}{5}x+2$ is dashed (due to $>$). Test $(0,0)$: $0 > \frac{-3}{5}(0)+2 \to 0 > 2$, false.

Step6: Shade Q15 valid half-plane

Shade the half-plane opposite $(0,0)$ (above the dashed line).

Step7: Analyze Q16 boundary & test point

Inequality: $y \geq -6$. Boundary line $y=-6$ is solid (due to $\geq$). Test $(0,0)$: $0 \geq -6$, true.

Step8: Shade Q16 valid half-plane

Shade the half-plane containing $(0,0)$ (above/on the solid line).

Answer:

1. For $y \leq 3x + 4$: Shade the half-plane below and including the solid line $y=3x+4$.
2. For $y < 7x - 2$: Shade the half-plane below the dashed line $y=7x-2$.
3. For $y > \frac{-3}{5}x + 2$: Shade the half-plane above the dashed line $y=\frac{-3}{5}x+2$.
4. For $y \geq -6$: Shade the half-plane above and including the solid horizontal line $y=-6$.