Question

which region should be shaded for the inequality $y < 3x - 1$?
a. no shading is needed
b. on the line
c. right of the line

in graphing $x + 4y \leq 8$, if the test point (1,1) does satisfy the inequality, where should you shade?
a. no shading is needed
b. below the line
c. on the line

Explanation:

Step1: Analyze first inequality

The inequality is $y < 3x - 1$. First, graph the boundary line $y=3x-1$ (dashed, since it's $<$ not $\leq$). For linear inequalities in this form, $y < \text{[linear expression]}$ means we shade the region below the line, which corresponds to the area right of the line for this positive-slope line.

Step2: Evaluate first question options

Among the given options, "Right of the line" matches the correct shaded region.

Step3: Analyze second inequality

The inequality is $x + 4y \leq 8$. We are told the test point $(1,1)$ satisfies it. Substitute to verify: $1 + 4(1) = 5 \leq 8$, which is true. The test point $(1,1)$ lies below the boundary line $x+4y=8$, so we shade the region containing this point.

Step4: Evaluate second question options

Among the given options, "Below the line" matches the correct shaded region.

Answer:

1. c. Right of the line
2. b. Below the line