# Graph a Linear Inequality in Two Variables

### Learning Outcome

- Identify and graph a linear inequality in two variables

- Graph the related boundary line. Replace the given inequality symbol, <, >, ≤ or ≥, in the inequality with the equality symbol, =, to find the equation of the boundary line.
- Identify at least one ordered pair on either side of the boundary line and substitute those [latex](x,y)[/latex] values into the inequality. Shade the region that contains the ordered pairs that make the inequality a true statement.
- If points on the boundary line are solutions, then use a solid line for drawing the boundary line. This will be the case for inequality with equality, ≤ or ≥.
- If points on the boundary line are not solutions, then use a dotted line for the boundary line. This will be the case for strict inequality, < or >.

*x*and

*y*-intercepts for this equation by substituting [latex]0[/latex] in for

*x*first and finding the value of

*y*; then substitute [latex]0[/latex] in for

*y*and find

*x*.

x |
y |

[latex]0[/latex] | [latex]1[/latex] |

[latex]4[/latex] | [latex]0[/latex] |

[latex]\begin{array}{r}−1+4\left(3\right)\leq4\\−1+12\leq4\\11\leq4\end{array}[/latex]

This is a false statement since [latex]11[/latex] is not less than or equal to [latex]4[/latex]. On the other hand, if you substitute [latex](2,0)[/latex] into [latex]x+4y\leq4[/latex]:[latex]\begin{array}{r}2+4\left(0\right)\leq4\\2+0\leq4\\2\leq4\end{array}[/latex]

This is true! The region that includes [latex](2,0)[/latex] should be shaded, as this is the region of solutions for the inequality. And there you have it, the graph of the set of solutions for [latex]x+4y\leq4[/latex]. Below is a video about how to graph inequalities with two variables. https://youtu.be/2VgFg2ztspI### Example

Graph the inequality [latex]2y>4x–6[/latex].Answer:
Solve for *y*.

[latex] \displaystyle \begin{array}{r}2y>4x-6\\\\\dfrac{2y}{2}>\dfrac{4x}{2}-\dfrac{6}{2}\\\\y>2x-3\\\end{array}[/latex]

Create a table of values to find two points on the line [latex] \displaystyle y=2x-3[/latex].x | y |
---|---|

[latex]0[/latex] | [latex]−3[/latex] |

[latex]2[/latex] | [latex]1[/latex] |

The inequality is [latex]2y>4x–6[/latex]. Find an ordered pair on either side of the boundary line. Insert the *x* and *y*-values into the inequality. If the simplified result is true, then shade on the side of the line the point is located.

[latex]\begin{array}{l}\\\text{Test }1:\left(−3,1\right)\\2\left(1\right)>4\left(−3\right)–6\\\,\,\,\,\,\,\,2>–12–6\\\,\,\,\,\,\,\,2>−18\\\,\,\,\,\,\,\,\,\text{TRUE}\\\\\text{Test }2:\left(4,1\right)\\2(1)>4\left(4\right)– 6\\\,\,\,\,\,\,2>16–6\\\,\,\,\,\,\,2>10\\\,\,\,\,\,\text{FALSE}\end{array}[/latex]

Since [latex](−3,1)[/latex] results in a true statement, the region that includes [latex](−3,1)[/latex] should be shaded.

The graph of the inequality [latex]2y>4x–6[/latex] is:Below is a video about how to graph inequalities with two variables when the equation is in slope-intercept form.

https://youtu.be/Hzxc4HASygU## Summary

When inequalities are graphed on a coordinate plane, the solutions are located in a region of the coordinate plane which is represented as a shaded area on the plane. The boundary line for the inequality is drawn as a solid line if the points on the line itself do satisfy the inequality, as in the cases of inequality with equality, [latex]\leq[/latex] and [latex]\geq[/latex]. It is drawn as a dashed line if the points on the line do not satisfy the inequality, as in the cases of strict inequality, < and >. You can tell which region to shade by testing some points in the inequality. Using a coordinate plane is especially helpful for visualizing the region of solutions for inequalities with two variables.## Licenses & Attributions

### CC licensed content, Original

- Ex 1: Graphing Linear Inequalities in Two Variables (Slope Intercept Form).
**Authored by:**James Sousa (Mathispower4u.com) .**License:**CC BY: Attribution.

### CC licensed content, Shared previously

- Ex 2: Graphing Linear Inequalities in Two Variables (Standard Form).
**Authored by:**James Sousa (Mathispower4u.com).**License:**CC BY: Attribution. -
**Provided by:**Monterey Institute of Technology**Located at:**https://www.nroc.org/.**License:**CC BY: Attribution.