Question

describing a system of two - variable inequalities
which statements are true about the graph of ( yleq3x + 1 ) and ( ygeq - x + 2 )? check all that apply.

• the slope of one boundary line is 2
• both boundary lines are solid
• a solution to the system is ( (1, 3) ).
• both inequalities are shaded below the boundary lines.
• 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 solidity

Both 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 line intersection

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

Answer:

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