Question

which is the graph of the system of inequalities $y \geq \frac{4}{5}x - \frac{1}{5}$ and $y \leq 2x + 6$?

Explanation:

Step1: Analyze \( y \geq \frac{4}{5}x - \frac{1}{5} \)

The inequality \( y \geq \frac{4}{5}x - \frac{1}{5} \) has a slope of \( \frac{4}{5} \) (positive, less steep) and a y - intercept of \( -\frac{1}{5} \). Since the inequality is \( \geq \), the line should be solid, and we shade above the line.

Step2: Analyze \( y \leq 2x + 6 \)

The inequality \( y \leq 2x + 6 \) has a slope of \( 2 \) (positive, steeper) and a y - intercept of \( 6 \). Since the inequality is \( \leq \), the line should be solid, and we shade below the line.

Step3: Compare with graphs

• The first two graphs have lines with positive slopes. The line with slope \( \frac{4}{5} \) is less steep than the line with slope \( 2 \). The region that satisfies both inequalities is where we are above the less - steep line (\( y\geq\frac{4}{5}x - \frac{1}{5} \)) and below the steeper line (\( y\leq2x + 6 \)).
• Looking at the four graphs, the fourth graph (the one with the two lines, one less steep with slope \( \frac{4}{5} \) and one steeper with slope \( 2 \), and the shaded region between them (above the less - steep line and below the steeper line)) is the correct one. (Assuming the fourth graph has the correct shading: above \( y = \frac{4}{5}x-\frac{1}{5} \) and below \( y = 2x + 6 \))

Answer:

The fourth graph (the one with the two lines, one with slope \( \frac{4}{5} \) and one with slope \( 2 \), and the shaded region that is above \( y=\frac{4}{5}x - \frac{1}{5} \) and below \( y = 2x+6 \))