Question

sketch the solution to each system of inequalities

1. $y < -dfrac{1}{3}x + 1$

$x \leq -5$

Explanation:

Step1: Identify boundary line

The inequality $y < -\frac{1}{3}x + 1$ has a boundary line $y = -\frac{1}{3}x + 1$. Since the inequality is strict ($<$), the line is dashed.

Step2: Find intercepts for boundary

For $x=0$, $y=1$ (y-intercept: $(0,1)$). For $y=0$, $0 = -\frac{1}{3}x + 1 \implies x=3$ (x-intercept: $(3,0)$). Plot these points and draw a dashed line through them.

Step3: Test a point for shading

Use the test point $(0,0)$: $0 < -\frac{1}{3}(0) + 1 \implies 0 < 1$, which is true. Shade the region below the dashed line.

Answer:

1. Draw a dashed line connecting $(0,1)$ and $(3,0)$.
2. Shade the entire region that lies below this dashed line. This shaded area represents all solutions to $y < -\frac{1}{3}x + 1$.