Question

1. choose the inequality whose solution set is represented by this graph.

a. x - 3y < 5
b. x - 3y ≤ 5
c. x - 3y > 5
d. x - 3y ≥ 5

Explanation:

Step1: Find the slope - intercept form of the line

The general form of a linear inequality is $y>mx + b$ (for a dashed line and the region above the line) or $y\frac{1}{3}x-\frac{5}{3}$. For $x - 3y>5$, we have $-3y>-x + 5$, then $y<\frac{1}{3}x-\frac{5}{3}$. For $x-3y\leq5$, we get $-3y\leq - x+5$, then $y\geq\frac{1}{3}x-\frac{5}{3}$. For $x - 3y\geq5$, we have $-3y\geq - x + 5$, then $y\leq\frac{1}{3}x-\frac{5}{3}$.
The line in the graph is dashed, so the inequality is either $<$ or $>$. The region is below the line, so the inequality is of the form $yRewrite the inequalities in slope - intercept form. Starting from $x-3y>5$, we solve for $y$:
\[

$$\begin{align*} x-3y&>5\\ -3y&>-x + 5\\ y&<\frac{1}{3}x-\frac{5}{3} \end{align*}$$

\]

Step2: Check the options

Option A: $x-3y<5$ can be rewritten as $y>\frac{1}{3}x-\frac{5}{3}$ which represents the region above the line. Option B: $x - 3y\leq5$ represents a solid - line inequality and the region above the line. Option C: $x-3y>5$ can be rewritten as $y<\frac{1}{3}x-\frac{5}{3}$ which represents the region below the line and has a dashed line. Option D: $x-3y\geq5$ represents a solid - line inequality and the region below the line.

Answer:

C. $x - 3y>5$