Question

which of the following inequalities is modeled by the graph?
○ $x + 4y \geq 15; y \geq 0$
○ $x - 4y \geq 15; y \geq 0$
○ $x + 4y \leq 15; y \geq 0$
○ $-x - 4y \geq 15; y \geq 0$

Explanation:

Step1: Find the equation of the line

The line passes through \((0, 3.75)\) (since when \(x = 0\), \(y = 3.75\)) and \((15, 0)\). The slope \(m=\frac{0 - 3.75}{15 - 0}=\frac{-3.75}{15}=-\frac{1}{4}\). Using the slope - intercept form \(y=mx + b\), with \(b = 3.75=\frac{15}{4}\), the equation of the line is \(y=-\frac{1}{4}x+\frac{15}{4}\). Multiply both sides by 4: \(4y=-x + 15\), which can be rewritten as \(x + 4y=15\).

Step2: Determine the inequality sign

The line is solid, and the shaded region is below the line (since the shading is towards the origin or the lower - left side relative to the line) and \(y\geq0\) (the shaded region is above \(y = 0\)). For a linear inequality \(Ax+By = C\), if the shaded region is below the line \(y=mx + b\) (when the line is written in slope - intercept form), and the line is included (solid line), the inequality is \(Ax + By\leq C\) (when \(B>0\)). Also, \(y\geq0\) is satisfied as the shaded region is above the \(x\) - axis. So the inequality is \(x + 4y\leq15\) and \(y\geq0\).

Step3: Eliminate other options

• For \(x + 4y\geq15\): The shaded region should be above the line, but our shaded region is below, so this is incorrect.
• For \(x - 4y\geq15\): Rewriting as \(y\leq\frac{x - 15}{4}\), the slope and intercept are different from our line, so this is incorrect.
• For \(-x - 4y\geq15\): Rewriting as \(y\leq-\frac{x + 15}{4}\), the slope and intercept are different from our line, so this is incorrect.

Answer:

C. \(x + 4y\leq15;y\geq0\) (assuming the options are labeled A, B, C, D with C being \(x + 4y\leq15;y\geq0\))