Question

1. which system of inequalities is represented by the graph?

$\bigcirc y < -x + 5$ and $y > \frac{2}{3}x + 2$
$\bigcirc y > -x + 5$ and $y > \frac{2}{3}x + 2$
$\bigcirc y > -x + 5$ and $y < \frac{2}{3}x + 2$
$\bigcirc y < -x + 5$ and $y < \frac{2}{3}x + 2$

Explanation:

Step1: Analyze the first line (slope -1, y - intercept 5)

The line \(y=-x + 5\) is dashed, so the inequality is either \(y>-x + 5\) or \(y<-x + 5\). The shaded region is above this line (since for a point above the line \(y=-x + 5\), say \((0,6)\), \(6>-0 + 5\) is true), so the inequality is \(y>-x + 5\).

Step2: Analyze the second line (slope \(\frac{2}{3}\), y - intercept 2)

The line \(y=\frac{2}{3}x+2\) is dashed, so the inequality is either \(y>\frac{2}{3}x + 2\) or \(y<\frac{2}{3}x+2\). The shaded region is above this line (for a point above the line, say \((0,3)\), \(3>\frac{2}{3}(0)+2\) is true), so the inequality is \(y>\frac{2}{3}x + 2\).

Answer:

\(y > -x + 5\) and \(y > \frac{2}{3}x + 2\) (the second option in the list: \(y > -x + 5\) and \(y > \frac{2}{3}x + 2\))