Question

graph the linear inequality
$x + 2y \geq -5$
use the graphing tool to graph the inequality
click to enlarge graph

Explanation:

Step1: Rewrite the inequality in slope - intercept form

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

Step2: Graph the boundary line

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

• To find the \(y\) - intercept: When \(x = 0\), \(y=-\frac{5}{2}=-2.5\). So the line passes through the point \((0,-2.5)\).
• To find the \(x\) - intercept: When \(y = 0\), we have \(0=-\frac{1}{2}x-\frac{5}{2}\). Add \(\frac{1}{2}x\) to both sides: \(\frac{1}{2}x=-\frac{5}{2}\). Multiply both sides by 2: \(x=- 5\). So the line passes through the point \((-5,0)\).

We can also use the slope \(m =-\frac{1}{2}\). From the \(y\) - intercept \((0,-2.5)\), we can go down 1 unit and to the right 2 units (or up 1 unit and to the left 2 units) to find other points on the line.

Step3: Shade the solution region

Since the inequality is \(y\geq-\frac{1}{2}x - \frac{5}{2}\), we shade the region above the boundary line. This is because for a given \(x\) - value, the \(y\) - values that satisfy \(y\geq-\frac{1}{2}x-\frac{5}{2}\) are the ones that are greater than or equal to the \(y\) - value on the line \(y =-\frac{1}{2}x-\frac{5}{2}\).

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

Answer:

The graph consists of a solid line \(y =-\frac{1}{2}x-\frac{5}{2}\) (passing through \((-5,0)\) and \((0, - 2.5)\)) and the region above the line is shaded.