Question

alities
$y \leq -\frac{2}{3}x - 1$
which of the following graphs represents the solution set of the inequality above?
four graphs labeled w, x, (and two others) with coordinate planes and shaded regions
session timer: 3:56

Explanation:

Step1: Analyze the inequality form

The inequality is \( y \leq -\frac{2}{3}x - 1 \). For a linear inequality \( y \leq mx + b \), the boundary line is \( y = mx + b \) (solid line because of \( \leq \), not \( < \)), and we shade below the line (since \( y \) is less than or equal to the line's value).

Step2: Identify the slope and y-intercept

The slope \( m = -\frac{2}{3} \) (negative, so the line decreases from left to right) and the y-intercept \( b = -1 \) (the line crosses the y-axis at \( (0, -1) \)).

Step3: Analyze each graph

• Graph W: Check the boundary line (solid? dashed? slope? y-intercept?) and shading. The boundary line here has a slope of \( -\frac{2}{3} \), y-intercept at \( (0, -1) \), and shading below the line (since \( y \leq \) the line). The line is solid (correct for \( \leq \)), slope and intercept match, and shading is below.
• Graph X: Shading seems above or incorrect, check intercept and slope.
• Other graphs: May have dashed lines (incorrect for \( \leq \)) or wrong shading direction or intercept.

Answer:

W (assuming the first graph labeled W is the correct one with solid line, slope \( -\frac{2}{3} \), y-intercept \( (0, -1) \), and shading below the line)