Question

which statements are true about the graph of the system of linear inequalities? select two options.
y > 3x - 4
y ≤ \\(\frac{1}{2}\\)x + 1
□ the graph of y > 3x - 4 has shading above a dashed line.
□ the graph of y ≤ \\(\frac{1}{2}\\)x + 1 has shading below a dashed line.
□ the graphs of the inequalities will intersect.
□ there are no solutions to the system.
□ the graphs of the two inequalities intersect the y - axis at (0, 1) and (0, 4).

Explanation:

Step1: Analyze $y>3x-4$

• Line type: Dashed (due to $>$), shading above the line (since $y$ is greater than the linear expression).

Step2: Analyze $y\leq\frac{1}{2}x+1$

• Line type: Solid (due to $\leq$), shading below the line (since $y$ is less than or equal to the linear expression).

Step3: Check intersection of graphs

Find intersection of $y=3x-4$ and $y=\frac{1}{2}x+1$:
Set $3x-4=\frac{1}{2}x+1$
$3x-\frac{1}{2}x=1+4$
$\frac{5}{2}x=5$
$x=2$, substitute back: $y=3(2)-4=2$. The lines intersect at $(2,2)$, so the shaded regions will intersect (there are solutions).

Step4: Check y-intercepts

For $y>3x-4$, y-intercept is $(0,-4)$. For $y\leq\frac{1}{2}x+1$, y-intercept is $(0,1)$.

Answer:

A. The graph of $y > 3x - 4$ has shading above a dashed line.
C. The graphs of the inequalities will intersect.