### Learning Outcome

- Identify whether an ordered pair is in the solution set of a linear inequality

The graph below shows the region of values that makes the inequality [latex]3x+2y\leq6[/latex] true (shaded red), the boundary line [latex]3x+2y=6[/latex], as well as a handful of ordered pairs. The boundary line is solid because points on the boundary line [latex]3x+2y=6[/latex] will make the inequality [latex]3x+2y\leq6[/latex] true.

You can substitute the *x* and *y-*values of each of the [latex](x,y)[/latex] ordered pairs into the inequality to find solutions. Sometimes making a table of values makes sense for more complicated inequalities.

Ordered Pair |
Makes the inequality [latex]3x+2y\leq6[/latex] a true statement |
Makes the inequality [latex]3x+2y\leq6[/latex] a false statement |
---|---|---|

[latex](−5, 5)[/latex] | [latex]\begin{array}{r}3\left(−5\right)+2\left(5\right)\leq6\\−15+10\leq6\\−5\leq6\end{array}[/latex] | |

[latex](−2,−2)[/latex] | [latex]\begin{array}{r}3\left(−2\right)+2\left(–2\right)\leq6\\−6+\left(−4\right)\leq6\\–10\leq6\end{array}[/latex] | |

[latex](2,3)[/latex] | [latex]\begin{array}{r}3\left(2\right)+2\left(3\right)\leq6\\6+6\leq6\\12\leq6\end{array}[/latex] | |

[latex](2,0)[/latex] | [latex]\begin{array}{r}3\left(2\right)+2\left(0\right)\leq6\\6+0\leq6\\6\leq6\end{array}[/latex] | |

[latex](4,−1)[/latex] | [latex]\begin{array}{r}3\left(4\right)+2\left(−1\right)\leq6\\12+\left(−2\right)\leq6\\10\leq6\end{array}[/latex] |

If substituting [latex](x,y)[/latex] into the inequality yields a true statement, then the ordered pair is a solution to the inequality, and the point will be plotted within the shaded region or the point will be part of a solid boundary line. A false statement means that the ordered pair is not a solution, and the point will graph outside the shaded region, or the point will be part of a dotted boundary line.

### Example

Use the graph to determine which ordered pairs plotted below are solutions of the inequality [latex]x–y<3[/latex].

The following video shows an example of determining whether an ordered pair is a solution to an inequality.

### Example

Is [latex](2,−3)[/latex] a solution of the inequality [latex]y<−3x+1[/latex]?

The following video shows another example of determining whether an ordered pair is a solution to an inequality.