Question

h the inequality on the axes below.

$3x - y \leq 7$

Explanation:

Step1: Convert to slope - intercept form

We start with the inequality \(3x - y\leq7\). To get it in the form \(y = mx + b\) (slope - intercept form), we solve for \(y\).
Subtract \(3x\) from both sides: \(-y\leq - 3x + 7\).
Multiply both sides by \(- 1\). Remember that when we multiply or divide an inequality by a negative number, the direction of the inequality sign changes. So we get \(y\geq3x - 7\).

Step2: Graph the boundary line

The boundary line is \(y = 3x-7\). This is a linear equation in slope - intercept form \(y=mx + b\), where the slope \(m = 3\) and the \(y\) - intercept \(b=-7\).

• Plot the \(y\) - intercept: The \(y\) - intercept is \((0,-7)\), so we plot the point \((0, - 7)\) on the \(y\) - axis.
• Use the slope to find another point: The slope \(m=\frac{rise}{run}=3=\frac{3}{1}\). From the point \((0,-7)\), we rise 3 units (move up 3) and run 1 unit (move to the right 1) to get the point \((1,-4)\). We can also go down 3 and left 1 from \((0,-7)\) to get \((- 1,-10)\).

Since the inequality is \(y\geq3x - 7\) (the "greater than or equal to" sign), the boundary line should be a solid line (because the points on the line are included in the solution set).

Step3: Shade the region

To determine which side of the line to shade, we can use a test point. A common test point is \((0,0)\) (as long as it is not on the boundary line).
Substitute \(x = 0\) and \(y = 0\) into the inequality \(y\geq3x - 7\):
\(0\geq3(0)-7\)
\(0\geq - 7\), which is a true statement.
So we shade the region that contains the point \((0,0)\). This region is above the line \(y = 3x-7\) (since \(y\) values are greater than or equal to the values on the line).

(Note: Since the problem asks to graph the inequality, the final answer is the graph with the solid line \(y = 3x - 7\) and the region above the line shaded. If we were to describe the steps for graphing, the above steps are the key steps. If we consider the process of graphing, the main steps are converting to slope - intercept form, graphing the boundary line, and shading the correct region.)

Answer:

The graph of the inequality \(3x - y\leq7\) (or \(y\geq3x - 7\)) has a solid line with slope \(3\) and \(y\) - intercept \(-7\), and the region above the line is shaded.