Question

graph the inequality on the axes below. \\(-3x + 4y \leq -20\\)

Explanation:

Step1: Rewrite the inequality in slope - intercept form

We start with the inequality \(-3x + 4y\leq - 20\).
First, we solve for \(y\). Add \(3x\) to both sides of the inequality:
\(4y\leq3x - 20\)
Then, divide each term by \(4\):
\(y\leq\frac{3}{4}x-5\)

Step2: Graph the boundary line

The boundary line is \(y = \frac{3}{4}x-5\). 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).
To find two points on the line, we can use the slope - intercept form. The \(y\) - intercept is \(- 5\), so one point is \((0,-5)\).
Using the slope \(\frac{3}{4}\) (rise over run), from the point \((0,-5)\), we move up \(3\) units and then \(4\) units to the right. So the next point is \((0 + 4,-5+3)=(4,-2)\).

Step3: Determine the region to shade

We test a point that is not on the line. A common test point is the origin \((0,0)\).
Substitute \(x = 0\) and \(y = 0\) into the inequality \(y\leq\frac{3}{4}x-5\):
\(0\leq\frac{3}{4}(0)-5\)
\(0\leq - 5\)
This is a false statement. So the origin is not in the solution set. We shade the region below the line \(y=\frac{3}{4}x - 5\) (since the inequality is \(y\leq\frac{3}{4}x-5\) and the origin is not in the solution set, we shade the side of the line that does not contain the origin).

(Note: Since this is a graphing problem, the final answer is the graph with a solid line \(y=\frac{3}{4}x - 5\) and the region below the line shaded. But in text - based form, we can describe the steps to graph it as above.)

Answer:

1. Rewrite the inequality as \(y\leq\frac{3}{4}x - 5\).
2. Graph the solid line \(y=\frac{3}{4}x - 5\) (using the \(y\) - intercept \((0,-5)\) and the point \((4,-2)\) from the slope).
3. Shade the region below the line (since the test point \((0,0)\) does not satisfy the inequality \(y\leq\frac{3}{4}x - 5\)).