Question

which statements are true about the graph of $y \leq 3x + 1$ and $y \geq -x + 2$? check all that apply. \
\square the slope of one boundary line is 2. \
\square both boundary lines are solid. \
\square a solution to the system is (1, 3). \
\square both inequalities are shaded below the boundary lines. \
\square the boundary lines intersect.

Explanation:

Step1: Check boundary line slopes

The boundary lines are $y=3x+1$ (slope $3$) and $y=-x+2$ (slope $-1$). No slope equals 2.

Step2: Check line type (solid/dashed)

Inequalities use $\leq$ and $\geq$, so lines are solid.

Step3: Test point (1,3)

For $y\leq3x+1$: $3\leq3(1)+1 \implies 3\leq4$, true. For $y\geq-x+2$: $3\geq-1+2 \implies 3\geq1$, true.

Step4: Check shading direction

$y\leq3x+1$ shades below; $y\geq-x+2$ shades above. Not both below.

Step5: Check if lines intersect

Slopes $3
eq-1$, so lines intersect.

Answer:

B. Both boundary lines are solid.
C. A solution to the system is (1, 3).
E. The boundary lines intersect.