# Solving Compound Inequalities

### Learning Outcomes

- Solve compound inequalities - OR - express solutions both graphically and with interval notation
- Solve compound inequalities - AND - express solutions both graphically and with interval notation

## Solve Compound Inequalities in the Form of *"or"*

As we saw in the last section, the solution of a compound inequality that consists of two inequalities joined with the word *or*is the union of the solutions of each inequality. Unions allow us to create a new set from two that may or may not have elements in common. In this section, you will see that some inequalities need to be simplified before their solution can be written or graphed. In the following example, you will see an example of how to solve a one-step inequality in the

*or*form. Note how each inequality is treated independently until the end, where the solution is described in terms of both inequalities. You will use the same properties to solve compound inequalities that you used to solve regular inequalities.

### Example

Solve for [latex]x[/latex]. [latex]3x–1<8[/latex]*or*[latex]x–5>0[/latex]

Answer: Solve each inequality by isolating the variable.

[latex] \displaystyle \begin{array}{r}x-5>0\,\,\,\,\,\,\,\,\textit{or}\,\,\,\,\,\,\,\,\,\,3x-1<8\,\,\\\underline{\,\,\,+5\,\,+5}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\,\,\,\,\,\,+1\,\,+1}\\x\,\,>5\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{3x}\,\,\,<\underline{9}\\{3}\,\,\,\,\,\,\,\,\,\,\,\,\,{3}\\x<3\,\,\,\\x>5\,\,\,\,\textit{or}\,\,\,\,x<3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex]

Inequality notation: [latex] \displaystyle x>5\,\,\,\textit{or}\,\,\,\,x<3[/latex] Interval notation: [latex]\left(-\infty, 3\right)\cup\left(5,\infty\right)[/latex] The solution to this compound inequality can also be shown graphically. Sometimes it helps to draw the graph first, before writing the solution using interval notation.### Example

Solve for [latex]y[/latex]. [latex]2y+7\lt13[/latex]*or*[latex]−3y–2\lt10[/latex]

Answer: Solve each inequality separately.

[latex] \displaystyle \begin{array}{r}2y+7<13\,\,\,\,\,\,\,\,\textit{or}\,\,\,\,\,\,\,\,\,\,-3y-2\lt 10\\\underline{\,\,\,-7\,\,-7}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\,\,\,\,\,\,+2\,\,\,+2}\\\underline{2y}<\underline{6}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{-3y}<\underline{12}\\{2}\,\,\,\,\,\,\,\,\,\,\,{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{-3}\,\,\,\,\,\,\,\,\,\,\,{-3}\\y<3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,y\gt -4\\y<3\,\,\,\,\textit{or}\,\,\,\,y\gt -4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex]

The inequality sign is reversed with division by a negative number. Since*y*could be less than [latex]3[/latex] or greater than [latex]−4[/latex],

*y*could be any number. Graphing the inequality helps with this interpretation. Inequality notation: [latex]y<3\text{ or }y> -4[/latex] Interval notation: [latex]\left(-\infty,\infty\right)[/latex] Graph: Even though the graph shows empty dots at [latex]y=3[/latex] and [latex]y=-4[/latex], they are included in the solution.

*y*.

### Example

Solve for [latex]z[/latex]. [latex-display]5z–3\gt−18[/latex]*or*[latex]−2z–1\gt15[/latex-display]

Answer: Solve each inequality separately. Combine the solutions.

[latex] \displaystyle \begin{array}{r}5z-3>18\,\,\,\,\,\,\,\,\textit{or}\,\,\,\,\,\,\,\,\,\,-2z-1>15\\\underline{\,\,\,+3\,\,\,+3}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\,\,\,\,\,\,+1\,\,\,+1}\\\underline{5z}>\underline{-15}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{-2z}>\underline{16}\\{5}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{5}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{-2}\,\,\,\,\,\,\,\,\,\,\,{-2}\\z>-3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,z<-8\\z>-3\,\,\,\,\textit{or}\,\,\,\,z<-8\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex]

Inequality notation: [latex] \displaystyle z>-3\,\,\,\,\textit{or}\,\,\,\,z<-8[/latex] Interval notation: [latex]\left(-\infty,-8\right)\cup\left(-3,\infty\right)[/latex] Note how we write the intervals with the one containing the most negative solutions first, then move to the right on the number line. [latex]z<-8[/latex] has solutions that continue all the way to the left on the number line, whereas [latex]x>-3[/latex] has solutions that continue all the way to the right. Graph:*or*and drawing the associated graph. https://youtu.be/oRlJ8G7trR8

### Try It

[ohm_question]3921[/ohm_question]## Solve Compound Inequalities in the Form of *"and"*

The solution of a compound inequality that consists of two inequalities joined with the word*and*is the intersection of the solutions of each inequality. In other words, both statements must be true at the same time. The solution to an

*and*compound inequality are all the solutions that the two inequalities have in common. As we saw in the last sections, this is where the two graphs overlap. In this section we will see more examples where we have to simplify the compound inequalities before we can express their solutions graphically or with an interval.

### Example

Solve for*x*. [latex] \displaystyle 1-4x\le 21\,\,\,\,\text{and}\,\,\,\,5x+2\ge22[/latex]

Answer:
Solve each inequality for *x*. Determine the intersection of the solutions.

[latex] \displaystyle \begin{array}{r}\,\,\,1-4x\le 21\,\,\,\,\,\,\,\,\text{AND}\,\,\,\,\,\,\,5x+2\ge 22\\\underline{-1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-1}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\,\,\,\,\,\,\,-2\,\,\,\,-2}\\\,\,\,\,\,\underline{-4x}\leq \underline{20}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{5x}\,\,\,\,\,\,\,\ge \underline{20}\\\,\,\,\,\,{-4}\,\,\,\,\,\,\,{-4}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{5}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{5}\,\,\\\,\,\,\,\,\,\,\,\,\,\,x\ge -5\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\ge 4\,\,\,\,\\\\x\ge -5\,\text{and}\,\,x\ge 4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex]

The number line below shows the graphs of the two inequalities in the problem. The solution to the compound inequality is [latex]x\geq4[/latex], since this is where the two graphs overlap.#### Answer

Inequality: [latex] \displaystyle x\ge 4[/latex] Interval: [latex]\left[4,\infty\right)[/latex] Graph:### EXample

Solve for*x*: [latex] \displaystyle {5}{x}-{2}\le{3}\text{ and }{4}{x}{+7}>{3}[/latex]

Answer: Solve each inequality separately. Find the overlap between the solutions.

[latex] \displaystyle \begin{array}{l}\,\,\,5x-2\le 3\,\,\,\,\,\,\,\,\,\text{AND}\,\,\,\,\,\,\,4x+7>\,\,\,\,3\\\underline{\,\,\,\,\,\,\,\,\,\,\,+2\,\,+2\,}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\,\,\,\,\,\,\,\,\,\,\,-7\,\,\,\,\,\,-7}\\\,\,\frac{5x}{5}\,\,\,\,\,\,\,\,\le \frac{5}{5}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{4x}{4}\,\,\,\,\,\,\,\,\,\,\,\,\,>\frac{-4}{4}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\le 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x>-1\\\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\le 1\,\,\,\,\text{and}\,\,\,\,x>-1\end{array}[/latex]

#### Answer

Inequality: [latex]-1\le{x}\le{1}[/latex] Interval: [latex]\left(-1,1\right)[/latex] Graph:### Try It

[ohm_question]3920[/ohm_question]## Compound inequalities in the form [latex]a<x<b[/latex]

Rather than splitting a compound inequality in the form of [latex]a<x<b[/latex] into two inequalities [latex]x<b[/latex]*and*[latex]x>a[/latex], you can more quickly to solve the inequality by applying the properties of inequality to all three segments of the compound inequality.

### Example

Solve for*x*. [latex]3\lt2x+3\leq 7[/latex]

Answer: Isolate the variable by subtracting [latex]3[/latex] from all [latex]3[/latex] parts of the inequality, then dividing each part by [latex]2[/latex].

[latex]\begin{array}{r}\,\,\,\,3\,\,\lt\,\,2x+3\,\,\leq \,\,\,\,7\\\underline{\,-3}\,\,\,\,\,\,\,\,\,\,\,\,\underline{\,\,\,\,\,\,\,-3}\,\,\,\,\,\,\,\,\underline{\,-3}\,\\\,\,\,\,\,\underline{\,0\,}\,\,\lt\,\,\,\,\underline{2x}\,\,\,\,\,\,\,\,\leq\,\,\,\underline{\,4\,}\\2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\,\,\\\,\,\,\,\,\,\,\,\,\,0\lt x\leq 2\,\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex]

#### Answer

Inequality: [latex] \displaystyle 0\lt{x}\le 2[/latex] Interval: [latex]\left(0,2\right][/latex] Graph:*x*. Whatever operation you perform on the middle portion of the inequality, you must also perform to each of the outside sections as well. Pay particular attention to division or multiplication by a negative. The solution to a compound inequality with

*and*is always the overlap between the solution to each inequality. There are three possible outcomes for compound inequalities joined by the word

*and*:

Case 1: | |
---|---|

Description | The solution could be all the values between two endpoints |

Inequalities | [latex]x\le{1}[/latex] and [latex]x\gt{-1}[/latex], or as a bounded inequality: [latex]{-1}\lt{x}\le{1}[/latex] |

Interval | [latex]\left(-1,1\right][/latex] |

Graphs | |

Case 2: | |

Description | The solution could begin at a point on the number line and extend in one direction. |

Inequalities | [latex]x\gt3[/latex] and [latex]x\ge4[/latex] |

Interval | [latex]\left[4,\infty\right)[/latex] |

Graphs | |

Case 3: | |

Description | In cases where there is no overlap between the two inequalities, there is no solution to the compound inequality |

Inequalities | [latex]x\lt{-3}[/latex] and [latex]x\gt{3}[/latex] |

Intervals | [latex]\left(-\infty,-3\right)[/latex] and [latex]\left(3,\infty\right)[/latex] |

Graph |

### Example

Solve for*x*. [latex]x+2>5[/latex] and [latex]x+4<5[/latex]

Answer: Solve each inequality separately.

[latex] \displaystyle \begin{array}{l}x+2>5\,\,\,\,\,\,\,\,\,\text{AND}\,\,\,\,\,\,\,x+4<5\,\,\,\,\\\underline{\,\,\,\,\,-2\,-2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\,\,\,\,\,\,-4\,-4}\\x\,\,\,\,\,\,\,\,>\,\,3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\,\,\,\,\,\,\,<\,1\\\\\,\,\,\,\,\,\,\,\,\,\,\,\,x>3\,\,\,\,\text{and}\,\,\,\,x<1\end{array}[/latex]

Find the overlap between the solutions.#### Answer

There is no overlap between [latex] \displaystyle x>3[/latex] and [latex]x<1[/latex], so there is no solution.## Summary

A compound inequality is a statement of two inequality statements linked together either by the word*or*or by the word

*and*. Sometimes, an

*and*compound inequality is shown symbolically, like [latex]a<x<b[/latex], and does not even need the word

*and*. Because compound inequalities represent either a union or intersection of the individual inequalities, graphing them on a number line can be a helpful way to see or check a solution. Compound inequalities can be manipulated and solved in much the same way any inequality is solved, by paying attention to the properties of inequalities and the rules for solving them.

## Contribute!

## Licenses & Attributions

### CC licensed content, Original

- Revision and Adaptation.
**Provided by:**Lumen Learning**License:**CC BY: Attribution.

### CC licensed content, Shared previously

- Ex 1: Solve a Compound Inequality Involving AND (Intersection).
**Authored by:**James Sousa (Mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution. - Unit 10: Solving Equations and Inequalities, from Developmental Math: An Open Program.
**Provided by:**Monterey Institute of Technology**Located at:**https://www.nroc.org/.**License:**CC BY: Attribution.