Question

identify the graph of $-x + 4y > -12$.

Explanation:

Step1: Rewrite the inequality in slope - intercept form

We start with the inequality \(-x + 4y>-12\).
To get it in the form \(y=mx + b\) (slope - intercept form), we solve for \(y\).
First, add \(x\) to both sides of the inequality: \(4y>x - 12\).
Then, divide each term by 4: \(y>\frac{1}{4}x-3\).

Step2: Analyze the boundary line

The boundary line for the inequality \(y>\frac{1}{4}x - 3\) is the line \(y = \frac{1}{4}x-3\). Since the inequality is \(y>\frac{1}{4}x - 3\) (not \(y\geq\frac{1}{4}x - 3\)), the boundary line should be a dashed line.
The slope of the line \(m=\frac{1}{4}\) and the \(y\) - intercept \(b=-3\).

Step3: Determine the region to shade

To determine which side of the line to shade, we can use a test point. A common test point is \((0,0)\) (as long as it is not on the boundary line).
Substitute \(x = 0\) and \(y = 0\) into the inequality \(y>\frac{1}{4}x-3\):
\(0>\frac{1}{4}(0)-3\)
\(0> - 3\), which is a true statement. So we shade the region that contains the point \((0,0)\).

Now, let's analyze the graphs:

• The boundary line \(y=\frac{1}{4}x - 3\) has a positive slope (\(\frac{1}{4}\)) and a \(y\) - intercept of \(-3\).
• Since \(y>\frac{1}{4}x - 3\), the region above the dashed line (because the inequality is "greater than") that contains \((0,0)\) should be shaded.

Looking at the four graphs, we check the position of the boundary line (slope and \(y\) - intercept) and the shaded region. The graph where the boundary line is dashed (since the inequality is strict, \(>\) not \(\geq\)), has a slope of \(\frac{1}{4}\), a \(y\) - intercept of \(-3\), and the region above the line (containing \((0,0)\)) is shaded is the correct one. From the given options, the bottom - right graph (the fourth one) has the correct boundary line (dashed, slope \(\frac{1}{4}\), \(y\) - intercept \(-3\)) and the correct shaded region (above the line, containing \((0,0)\)).

Answer:

The correct graph is the bottom - right graph (the one with the dashed line \(y = \frac{1}{4}x-3\) and the region above the line shaded, containing the point \((0,0)\)).