Question

solving a system of linear inequalities
which is the graph of the system of inequalities ( y geq \frac{4}{3}x - \frac{1}{3} ) and ( y leq 2x + 6 )

Explanation:

Step1: Analyze first inequality

The inequality is $y \geq \frac{4}{5}x - \frac{1}{5}$. This is a linear inequality with a positive slope $\frac{4}{5}$, y-intercept $-\frac{1}{5}$. The boundary line is solid, and we shade above the line (since $y$ is greater than or equal to the expression).

Step2: Analyze second inequality

The inequality is $y \leq 2x + 6$. This is a linear inequality with a positive slope $2$, y-intercept $6$. The boundary line is solid, and we shade below the line (since $y$ is less than or equal to the expression).

Step3: Match to correct graph

We need a graph with two solid lines: one with a gentle positive slope ($\frac{4}{5}$) shaded above, and one with a steeper positive slope ($2$) shaded below. The overlapping shaded region is the solution set, which matches the fourth graph (rightmost one).

Answer:

The correct graph is the rightmost option (the fourth one, with the overlapping shaded region between the two upward-sloping solid lines, above the gentler line and below the steeper line).