Question

graph the linear inequality.
x + 2y ≥ -10

use the graphing tool to graph the inequality.

Explanation:

Step1: Rewrite the inequality in slope - intercept form

We start with the inequality \(x + 2y\geq - 10\). To get it in the form \(y=mx + b\) (slope - intercept form), we solve for \(y\).
Subtract \(x\) from both sides: \(2y\geq -x - 10\).
Then divide each term by 2: \(y\geq-\frac{1}{2}x - 5\).

Step2: Graph the boundary line

The boundary line is \(y =-\frac{1}{2}x - 5\). This is a straight line.

• Find the y - intercept: When \(x = 0\), \(y=-5\). So the line passes through the point \((0,-5)\).
• Find the x - intercept: When \(y = 0\), \(0=-\frac{1}{2}x-5\). Add 5 to both sides: \(\frac{1}{2}x=- 5\), then multiply both sides by 2: \(x=-10\). So the line passes through the point \((-10,0)\).

Since the inequality is \(\geq\), the boundary line should be a solid line (because the points on the line are included in the solution set).

Step3: Determine the region to shade

We can use a test point that is not on the line. A common test point is \((0,0)\).
Substitute \(x = 0\) and \(y = 0\) into the original inequality \(x + 2y\geq - 10\):
\(0+2(0)\geq - 10\), which simplifies to \(0\geq - 10\). This is a true statement.
So we shade the region that contains the point \((0,0)\) (the region above the line \(y =-\frac{1}{2}x - 5\) since the test point \((0,0)\) is above the line \(y=-\frac{1}{2}(0)-5=-5\)).

(Note: Since this is a graphing problem, the final answer is the graph of the inequality \(x + 2y\geq - 10\) with a solid line \(y =-\frac{1}{2}x - 5\) and the region above the line shaded.)

Answer:

The graph of the inequality \(x + 2y\geq - 10\) has a solid line \(y=-\frac{1}{2}x - 5\) (passing through \((0, - 5)\) and \((-10,0)\)) and the region above the line is shaded.