Question

this is the graph of a linear inequality. write the inequality in slope - intercept form.
write your answer with y first, followed by an inequality symbol. use integers, proper fractions, and improper fractions in simplest form.

Explanation:

Step1: Find the slope and y-intercept of the boundary line

The boundary line is a dashed line (so the inequality is strict, either \(>\) or \(<\)) and passes through the points \((0, -2)\) (y-intercept) and \((4, -1)\) (we can find another point by moving 4 units right and 1 unit up from the y-intercept). The slope \(m\) is calculated as \(\frac{y_2 - y_1}{x_2 - x_1}=\frac{-1 - (-2)}{4 - 0}=\frac{1}{4}\). The y-intercept \(b\) is \(-2\). So the equation of the boundary line in slope - intercept form (\(y = mx + b\)) is \(y=\frac{1}{4}x-2\).

Step2: Determine the inequality symbol

The shaded region is above the boundary line (we can test a point in the shaded region, for example, \((0,0)\). Substitute into the inequality: \(0\)? \(\frac{1}{4}(0)-2\). \(0\) is greater than \(- 2\). Since the boundary line is dashed, the inequality is \(y>\frac{1}{4}x - 2\).

Answer:

\(y>\frac{1}{4}x - 2\)