Determine 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.

\(\left\{\begin{array}{l}x+4 y \geq 10 \\ 3 x-2 y < 12\end{array}\right.\)

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 \((x,y)\) 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 5.51

Determine whether the ordered pair is a solution to the system.

\(\left\{\begin{array}{l}x+4 y \geq 10 \\ 3 x-2 y < 12\end{array}\right.\)

  1. (−2,4) 
  2. (3,1)
Solution
  1. Is the ordered pair (−2, 4) a solution?

    We substitute \(x=-2\) and \(y=4\) into both inequalities.

    \(\begin{array}{rlrl}x+4 & \geq 10 & 3 x-2 & < 12 \\ -2+4(4) & \stackrel{?}{\geq} & 3(-2)-2(4) & \stackrel{?}{ < } 12 \\ 14 & \geq 10 \text { true } & -14 & < 12 \, \text{true} \end{array}\)

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

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

    We substitute \(x=3\) and \(y=1\) into both inequalities.

    \(\begin{array}{rrr}x+4 \geq 10 & \qquad \qquad 3 x-2 < 12 \\ 3+4(1) \geq 10 & 3(3)-2(1) \stackrel{?}{ < } 12 \\ 7 \geq 10 \text { false } & \qquad 7 < 12 \text { true }\end{array}\)

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

Try It 5.101

Determine whether the ordered pair is a solution to the system.

\(\left\{\begin{array}{l}x-5 y > 10 \\ 2 x+3 y > -2\end{array}\right.\)

  1. \((3,−1)\)
  2. \((6,−3)\)
Try It 5.102

Determine whether the ordered pair is a solution to the system.
\(\left\{\begin{array}{l}y > 4 x-2 \\ 4 x-y < 20\end{array}\right.\)

  1. \((2,1)\)
  2. \((4,−1)\)
Back to '6.7: Graphing Systems of Linear Inequalities'