Determining Whether An Ordered Pair is a Solution of a System of Linear Inequalities

The definition of a system of linear inequalities is very similar to the definition of a system of linear equations.

System of Linear Inequalities

Two or more linear inequalities grouped together form a system of linear inequalities.

A system of linear inequalities looks like a system of linear equations, but it has inequalities instead of equations. A system of two linear inequalities is shown below.

\(\begin{array}{c}x+4y\ge 10\hfill \\ 3x-2y<12\hfill \end{array}\)

To solve a system of linear inequalities, we will find values of the variables that are solutions to both inequalities. We solve the system by using the graphs of each inequality and show the solution as a graph. We will find the region on the plane that contains all ordered pairs \(\left(x,y\right)\) that make both inequalities true.

Solutions of a System of Linear Inequalities

Solutions of a system of linear inequalities are the values of the variables that make all the inequalities true.

The solution of a system of linear inequalities is shown as a shaded region in the x-y coordinate system that includes all the points whose ordered pairs make the inequalities true.

To determine if an ordered pair is a solution to a system of two inequalities, we substitute the values of the variables into each inequality. If the ordered pair makes both inequalities true, it is a solution to the system.

Example

Determine whether the ordered pair is a solution to the system. \(\begin{array}{c}x+4y\ge 10\hfill \\ 3x-2y<12\hfill \end{array}\)

(a) (−2, 4)
(b) (3,1)

Solution

(a) Is the ordered pair (−2, 4) a solution?

This figure says, “We substitute x = -2 and y = 4 into both inequalities. The first inequality, x + 4 y is greater than or equal to 10 becomes -2 plus 4 times 4 is greater than or less than 10 or 14 is great than or less than 10 which is true. The second inequality, 3x – 2y is less than 12 becomes 3 times -2 – 2 times 4 is less than 12 or -14 is less than 12 which is true.

The ordered pair (−2, 4) made both inequalities true. Therefore (−2, 4) is a solution to this system.

(b) Is the ordered pair (3,1) a solution?

This figure says, “We substitute x 3 and y = 1 into both inequalities.” The first inequality, x + 4y is greater than or equal to 10 becomes 3 + 4 times 1 is greater than or equal to 10 or y is greater than or equal to 10 which is false. The second inequality, 3x -2y is less than 12 becomes 3 times 3 – two times 1 is less than 12 or 7 is less than 12 which is true.

The ordered pair (3,1) made one inequality true, but the other one false. Therefore (3,1) is not a solution to this system.

Determining Whether An Ordered Pair is a Solution of a System of Linear Inequalities