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Study Guides > Intermediate Algebra

Solve Compound Inequalities—OR

Learning Outcome

  • Solve compound inequalities in the form of or and express the solution 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 x.  [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. Number line. Open red circle on 3 and red highlight through all numbers less than 3. Open blue circle on 5 and blue highlight on all numbers greater than 5.

Remember to apply the properties of inequalities when you are solving compound inequalities. The next example involves dividing by a negative to isolate a variable.

Example

Solve for y.  [latex]2y+7\lt13\textit{ or }−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: Open dot on negative 4 and shaded line going through all numbers greater than negative 4. Open dot on 3 and shaded line on all numbers less than 3. Numbers between closed dot on negative 4 and open dot on 3 are shaded twice. Even though the graph shows empty dots at [latex]y=3[/latex] and [latex]y=-4[/latex], they are included in the solution.

In the last example, the final answer included solutions whose intervals overlapped. This caused the answer to include all the numbers on the number line. In words, we call this solution "all real numbers."  Any real number will produce a true statement for either [latex]y<3\text{ or }y\gt -4[/latex] when it is substituted for 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:Number line. Red open circle on negative 8 and red highlight on all numbers less than negative 8. Open blue circle on negative 3 and blue highlight through all numbers greater than negative 3.

The following video contains an example of solving a compound inequality involving or and drawing the associated graph. https://youtu.be/oRlJ8G7trR8 In the next section you will see examples of how to solve compound inequalities containing and.

Licenses & Attributions

CC licensed content, Original

CC licensed content, Shared previously

  • Ex: Solve a Compound Inequality Involving OR (Union). 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 and Education Located at: https://www.nroc.org/. License: CC BY: Attribution.