Question

graph the inequality.
$x + 4y \leq -4$

Explanation:

Answer:

To graph \( x + 4y \leq -4 \), follow these steps:

Step 1: Rewrite in slope - intercept form

First, solve the inequality for \( y \).
Subtract \( x \) from both sides: \( 4y\leq -x - 4 \)
Divide each term by 4: \( y\leq-\frac{1}{4}x - 1 \)

Step 2: Graph the boundary line

The boundary line is \( y =-\frac{1}{4}x - 1 \).

• Find the y - intercept: When \( x = 0 \), \( y=-1 \). So the line passes through the point \( (0,-1) \).
• Find the x - intercept: When \( y = 0 \), \( 0=-\frac{1}{4}x - 1 \). Add 1 to both sides: \( 1=-\frac{1}{4}x \). Multiply both sides by - 4: \( x=-4 \). So the line passes through the point \( (-4,0) \).
• Since the inequality is \( \leq \), the boundary line should be a solid line (because the points on the line are included in the solution set).

Step 3: Shade the region

To determine which side of the line to shade, we can use a test point. A common test point is \( (0,0) \).
Substitute \( x = 0 \) and \( y = 0 \) into the original inequality \( x + 4y\leq - 4 \):
\( 0+4(0)\leq - 4 \)
\( 0\leq - 4 \), which is false.
So we shade the region that does not contain the point \( (0,0) \). That is, the region below the line \( y =-\frac{1}{4}x - 1 \) (since the test point above the line gave a false statement, we shade the opposite side).

The graph will have a solid line passing through \( (-4,0) \) and \( (0,-1) \), with the area below the line shaded.