Solve systems of linear inequalities graphically

Detailed Explanation: Solve systems of linear inequalities graphically

A system of linear inequalities asks for all points that satisfy both inequalities at the same time. Graph each inequality, shade its solution region, and identify the overlapping shaded region.

Worked example

Solve graphically:

1. Graph each boundary line

For the first inequality, graph the line

Its slope is $2$ and its $y$-intercept is $-1$. Because the inequality is $\ge$, draw this boundary as a solid line. The line itself is included.

For the second inequality, graph the line

Its slope is $-1$ and its $y$-intercept is $5$. Because the inequality is $<$, draw this boundary as a dashed line. The line itself is not included.

The two lines meet at $(2,3)$ because

gives $x=2$ and then $y=3$.

2. Shade the region for each inequality

For

shade above the solid line because $y$ is greater than or equal to the expression.

For

shade below the dashed line because $y$ is less than the expression.

3. Find the overlap

The solution is the region that is shaded both above the first line and below the second line.

A point such as $(0,0)$ helps verify the overlap:

• $0\ge 2(0)-1$, so $0\ge -1$ is true.
• $0<-0+5$, so $0<5$ is true.

Therefore, $(0,0)$ is in the solution region.

The overlapping region lies between the two lines, extending to the left. The point $(2,3)$ is not included because it lies on the dashed boundary $y=-x+5$.

Thus, the graphical solution is the region

with the first boundary solid and the second boundary dashed.