Question

which statements are true about the graph of the system of linear inequalities? select two options.
y > 3x - 4
y ≤ (1/2)x + 1
□ the graph of y > 3x - 4 has shading above a dashed line.
□ the graph of y ≤ (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 \( y > 3x - 4 \)

For a linear inequality \( y > mx + b \), the line is dashed (since the inequality is strict, \( > \) not \( \geq \)) and shading is above the line (because \( y \) values greater than \( 3x - 4 \) are above the line). So the first statement is true.

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

For \( y \leq \frac{1}{2}x + 1 \), the line is solid (because of \( \leq \), not strict) and shading is below the line. The second statement says dashed line, which is wrong.

Step3: Check intersection of inequalities

The first line has slope 3, the second has slope \( \frac{1}{2} \). Different slopes mean the lines intersect, so their solution regions (shaded areas) will also intersect (since one is above a dashed line and one is below a solid line, and lines intersect). So the third statement is true.

Step4: Check y - intercepts

For \( y > 3x - 4 \), y - intercept is \( (0, - 4) \). For \( y \leq \frac{1}{2}x + 1 \), y - intercept is \( (0, 1) \). The fifth statement says \( (0, 4) \) for the first, which is wrong. The fourth statement says no solutions, but we know they intersect, so wrong.

Answer:

The graph of \( y > 3x - 4 \) has shading above a dashed line.
The graphs of the inequalities will intersect.