Question

Graph the system of linear inequalities. $$y>-2 x+2\( \)y \leq-1$$

Short answer

Graph the two inequalities by creating dashed or solid lines based on the inequality symbol, then shade the correct side of the lines. The solution of the system of inequalities is where the shaded regions overlap.

Step-by-step solution

Graph the first inequality

The first inequality is \(y > -2x + 2\). This is in the slope-intercept form of a linear equation, so it can be seen that the y-intercept is 2 and the slope is -2. First graph the line \(y = -2x + 2\), but since the inequality is 'greater than' and not 'greater than or equal to', we will represent this line as a dashed line, to indicate that the points on the line are not part of the solution. Then choose a test point not on the line, such as the origin (0,0), to determine which side of the line to shade. Substitute (0,0) into the inequality, if it makes the inequality true, then shade that side of the line. In this case, (0,0) does not satisfy the inequality so we shade the side of the line that does not include the origin.

Graph the second inequality

The second inequality is \(y \leq -1\). This is a horizontal line denoted by y = -1. Unlike the first inequality, this line is solid because the inequality sign is 'less than or equal to', meaning that points on the line are included in the solution set. Graph the line, and once again, choose a test point, such as the origin, to determine which side to shade. In this case, (0,0) does not satisfy the inequality, so we shade the side of the line that does not include the origin.

Find the intersection of the two regions

The solution to this system of inequalities is the intersection of the areas that satisfy both inequalities. This intersection region represents all points that satisfy both inequalities.