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Write an inequality to represent the graph shown.
The slope of the boundary is \(-\frac{1}{3}\), so the equation of the boundary will be of the form \(y = -\frac{1}{3}x + b\). The \(y\)-intercept is not clear, so use another point on the boundary to determine \(b\).
The equation of the boundary is \(y = - \frac{1}{3}x + \frac{4}{3}\). The boundary is a solid line, so the inequality is not strict. The lower portion of the graph is shaded, so the inequality must be of the form \(y \le\) ____. An inequality that corresponds to the graph is \(y \le - \frac{1}{3}x + \frac{4}{3}\).
Alternatively, use a test point from the solution region to determine the direction of the inequality symbol. The point \((0, 0)\) is a point in the solution region.
An inequality that corresponds to the graph is \(y \le - \frac{1}{3}x + \frac{4}{3}\).
\[\begin{align}
y &= - \frac{1}{3}x + b \\
0 &= - \frac{1}{3}\left( 4 \right) + b \\
0 &= - \frac{4}{3} + b \\
\frac{4}{3} &= b \\
\end{align}\]
y &= - \frac{1}{3}x + b \\
0 &= - \frac{1}{3}\left( 4 \right) + b \\
0 &= - \frac{4}{3} + b \\
\frac{4}{3} &= b \\
\end{align}\]
The equation of the boundary is \(y = - \frac{1}{3}x + \frac{4}{3}\). The boundary is a solid line, so the inequality is not strict. The lower portion of the graph is shaded, so the inequality must be of the form \(y \le\) ____. An inequality that corresponds to the graph is \(y \le - \frac{1}{3}x + \frac{4}{3}\).
Alternatively, use a test point from the solution region to determine the direction of the inequality symbol. The point \((0, 0)\) is a point in the solution region.
| Left Side | Right Side |
|---|---|
|
\[\begin{array}{r} y \\ 0 \end{array}\] |
\(\begin{array}{l} -\frac{1}{3}x + \frac{4}{3} \\ -\frac{1}{3}(0) + \frac{4}{3} \\ \frac{4}{3} \end{array}\) |
| \(\hspace{25pt}\)LS \(\le \) RS | |
An inequality that corresponds to the graph is \(y \le - \frac{1}{3}x + \frac{4}{3}\).