Question

1. $-x + y > -2$ $x + 2y < 2$ to the teacher: for online: this assessment contains question(s) using the drawing tool. you may want to review the use of the drawing tool with your students prior to taking the test. an example question is given in the abeka sample test. for the pdf: this assessment contains question(s) involving graphing. you may want to edit the directions for that section since they currently apply to using the drawing tool in the online test. graph the system of inequalities. click the paintbrush button below the plane. use the tools to draw the line(s) and shading. click the green check mark to save your work. close

Explanation:

Step1: Rewrite the first inequality

Rewrite \(-x + y > -2\) in slope - intercept form (\(y=mx + b\)). Add \(x\) to both sides: \(y>x - 2\). The boundary line is \(y = x-2\) (dashed line because the inequality is strict, \(>\)), with a slope of \(1\) and a \(y\) - intercept of \(-2\). To graph the boundary line, plot the \(y\) - intercept \((0,-2)\), then use the slope (rise \(1\), run \(1\)) to find another point, e.g., \((1,-1)\). Then, shade the region above the line (since \(y>x - 2\)).

Step2: Rewrite the second inequality

Rewrite \(x + 2y<2\) in slope - intercept form. Subtract \(x\) from both sides: \(2y<-x + 2\). Divide both sides by \(2\): \(y<-\frac{1}{2}x + 1\). The boundary line is \(y=-\frac{1}{2}x + 1\) (dashed line because the inequality is strict, \(<\)), with a slope of \(-\frac{1}{2}\) and a \(y\) - intercept of \(1\). Plot the \(y\) - intercept \((0,1)\), then use the slope (rise \(- 1\), run \(2\)) to find another point, e.g., \((2,0)\). Then, shade the region below the line (since \(y<-\frac{1}{2}x + 1\)).

Step3: Find the solution region

The solution to the system of inequalities is the region where the two shaded regions overlap.

Answer:

The graph consists of two dashed lines: \(y = x - 2\) (with shading above) and \(y=-\frac{1}{2}x + 1\) (with shading below), and the overlapping shaded region is the solution to the system \(

$$\begin{cases}-x + y>-2\\x + 2y<2\end{cases}$$

\)