Question

Graph the system of linear inequalities. \(y>x\) \(x \leq 1\)

Short answer

The graph should show two lines; a dashed positively-sloped line and a solid vertical line at x=1. The area shaded on both the left side of x=1 and above y=x is our solution.

Step-by-step solution

Graph the first inequality

To graph \(y > x\), start by treating the inequality as \(y = x\) and draw that line. Since it is a 'greater than' not 'greater than or equal to', we will do it with a dashed line, indicating the line is not part of the solution. After drawing the line, we choose either side of the line and test a point to see if it satisfies the inequality. For instance, we can test the point (0,1). Since \(1 > 0\) is true, we realize that we will shade the area above the line.

Graph the second inequality

To graph \(x \leq 1\), again start by treating the inequality as \(x = 1\) and draw the line x=1. Because it is 'less than or equal to', we will draw a solid line to include all the x-values on the line. Then test any point to the left of the line and see if it satisfies the inequality. For instance, if we use the origin (0,0), we see that 0 is less than 1, so we would shade the area to the left of the line.

Complete the graph

Now that we have two shaded regions from each inequality, we look for the shared shaded region between both inequalities. This region is our solution to this system of inequalities.