Question

which of the following linear inequalities matches the graph above?
a ( y > \frac{3}{5}x + 7 )
b ( y < \frac{5}{3}x + 7 )
c ( y geq \frac{3}{5}x + 7 )
d ( y > -\frac{3}{5}x + 7 )

Explanation:

Step1: Analyze the line's slope and intercept

The line in the graph has a positive slope. Let's find the slope using two points. From the graph, when \(x = 0\), \(y = 7\) (y-intercept \(b = 7\)). Another point: when \(x = 5\), \(y = 10\)? Wait, no, let's calculate slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's take two points on the line: (0,7) and (5, 10)? Wait, no, maybe ( - 5, 4) and (0,7). Then \(m=\frac{7 - 4}{0 - (-5)}=\frac{3}{5}\). So the equation of the line is \(y=\frac{3}{5}x + 7\).

Step2: Determine the inequality sign and line type

The line is dashed (since the options have >, <, not ≥ or ≤? Wait, no, the options: A is \(y>\frac{3}{5}x + 7\), B is \(y<\frac{5}{3}x + 7\) (slope wrong), C is \(y\geq-\frac{3}{5}x + 7\) (slope negative, wrong), D is \(y>-\frac{3}{5}x + 7\) (slope negative, wrong). Wait, the shaded region is above the line? Wait, the line has positive slope \(\frac{3}{5}\), and the shaded area is to the right and above? Wait, no, let's check the slope again. Wait, maybe I made a mistake. Let's see the graph: the line goes from (0,7) and (5, 10)? Wait, no, maybe (0,7) and (5, 10) would have slope 3/5. The shaded region is above the line? Wait, the line is dashed (since the inequality is strict, > or <). The correct slope is 3/5, and the shaded area is above the line, so \(y>\frac{3}{5}x + 7\), which is option A. Wait, but let's check the options again. Option A: \(y>\frac{3}{5}x + 7\), option B: \(y<\frac{5}{3}x + 7\) (slope 5/3 is wrong), option C: \(y\geq-\frac{3}{5}x + 7\) (slope negative, wrong), option D: \(y>-\frac{3}{5}x + 7\) (slope negative, wrong). So the correct line has slope 3/5, equation \(y=\frac{3}{5}x + 7\), and the shaded region is above the line (so >) and the line is dashed (so strict inequality). So option A is correct.

Answer:

A. \( y > \frac{3}{5}x + 7 \)