Solving Systems of Inequalities

Solving Systems of Linear Inequalities in Two Variables

The solution of a system of linear inequalities is represented on a graph as the overlap of the shaded regions that represent the solutions of the separate inequalities.
A system of inequalities is a set of inequalities that are used at the same time. A solution of a system of inequalities is a set of values of the variables in the system that make every inequality in the system true. The solution set of a system of inequalities is often an infinite set of points. The solution set can be represented by graphing the inequalities in the system on the same coordinate plane and identifying the region where their solutions overlap. In some instances, a system of inequalities may have no solution. On a graph of a system of inequalities with no solution, the solutions of the inequalities in the system do not overlap.
Step-By-Step Example
Graphing a System of Linear Inequalities
Solve the system of inequalities.
{yx+85xy<4\begin{cases}\begin{aligned}y &\leq x+8 \\ 5x-y &< -4 \end{aligned}\end{cases}
Step 1

Solve the inequalities for yy .

The first inequality is already solved for yy.
yx+8y \leq x+8
Solve the second inequality for yy. When multiplying both sides of an inequality by a negative number, remember to reverse the inequality symbol.
5xy<45x+5xy<5x4y<5x41(y)>1(5x4)y>5x+4\begin{aligned}5x-y&<-4\\-5x+5x-y&<-5x-4\\-y&<-5x-4\\-1(-y)&>-1(-5x-4)\\y&>5x+4\end{aligned}
Step 2

Graph the boundary lines.

Use a solid line for:
yx+8y \leq x+8
Use a dashed line for:
y>5x+4y>5x+4
Solution

The inequality symbol in yx+8y \leq x+8 is \leq, so shade below the boundary line.

The inequality symbol in y>5x+4 y > 5x+4 is >\gt, so shade above the boundary line.

The solution is the region where the two shaded areas overlap.
Check the solution by selecting a test point from the shaded region that is not on either boundary line. Then substitute the coordinates into both inequalities to verify that each is true.

The point (0,6)(0, 6) is in the overlap of the shaded areas and is not on either line.

First inequality:
yx+8(6)?(0)+868\begin{aligned}y&\leq x + 8\\(6)&\stackrel{?}{\leq} (0)+8\\6&\leq 8 & \checkmark\end{aligned}
Second inequality:
5xy<45(0)(6)<?406<?46<4\begin{aligned}5x-y&<-4\\5(0)-(6)&\stackrel{?}{<}-4\\0-6&\stackrel{?}{<}-4\\-6&<-4 & \checkmark\end{aligned}
Both inequalities are true, so the region that contains (0,6)(0, 6) is the solution.
Step-By-Step Example
Graphing a System of Linear Inequalities with No Solution
Solve the system of inequalities.
{y<4x44xy6\begin{cases}\begin{aligned}y &< 4x-4 \\ 4x-y &\leq -6 \end{aligned}\end{cases}
Step 1

Solve the inequalities for yy.

The first inequality is already solved for yy.
y<4x4y<4x-4
Solve the second inequality for yy. When multiplying both sides of an inequality by a negative number, remember to reverse the inequality symbol.
4xy64x+4xy4x6y4x61(y)1(4x6)y4x+6\begin{aligned}4x-y&\leq -6\\-4x+4x-y&\leq -4x-6\\-y&\leq -4x-6\\-1(-y)&\geq -1(-4x-6)\\y&\geq 4x+6\end{aligned}
Step 2

Graph the boundary lines. The lines have the same slope but different yy-intercepts. This means that they are parallel, so they do not intersect.

Use a dashed line for y<4x4y \lt 4x-4.

Use a solid line for y4x+6y \geq 4x+6.
Solution

The inequality symbol in y<4x4y < 4x-4 is <\lt, so shade below the boundary line.

The inequality symbol in y4x+6y \geq 4x+6 is \geq, so shade above the boundary line.

The shaded areas do not overlap, so there is no solution.