Question

which linear inequality is represented by the graph?
$y \leq \frac{1}{3}x - 1$
$y \geq \frac{1}{3}x - 1$
$y < 3x - 1$
$y > 3x - 1$

Explanation:

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

The boundary line passes through points (0, -1) and (3, 0). The slope \( m \) is calculated as \( \frac{0 - (-1)}{3 - 0} = \frac{1}{3} \). The y-intercept \( b \) is -1 (since it crosses the y-axis at (0, -1)). So the equation of the boundary line is \( y = \frac{1}{3}x - 1 \).

Step2: Determine the inequality symbol (dashed or solid line, and shading)

The boundary line is solid (so the inequality includes equality, \( \leq \) or \( \geq \)). The shading is above the line. For a linear inequality, if the shading is above the line \( y = mx + b \), the inequality is \( y \geq mx + b \).

Step3: Eliminate incorrect options

• The slope is \( \frac{1}{3} \), not 3, so options with \( 3x - 1 \) (third and fourth) are eliminated.
• Shading is above the line, so we need \( y \geq \frac{1}{3}x - 1 \) (second option), not \( y \leq \frac{1}{3}x - 1 \) (first option).

Answer:

B. \( y \geq \frac{1}{3}x - 1 \)