Many times we encounter situations with multiple requirements. Sometimes we need only satisfy at least one requirement on a list, and other times we need to satisfy all the requirements on a list. For example, one class may require that you purchase a physical textbook or an e-text. If you have one of these, or both, you have satisfied the requirement. On the other hand, if your class requires you to have a textbook and a calculator, you must have both of these items to satisfy the requirement.

A compound inequality consists of two inequalities joined with the word and or the word or.

Solve Compound Inequalities in the Form of “or”

A compound inequality consisting of two inequalities joined with the word or has a solution set made up of all real numbers which satisfy the first inequality, the second inequality, or both. Unions allow us to create a new set from two that may or may not have elements in common.

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]

Show Solution

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.

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 [latex]y[/latex].

[latex]2y+7\lt13[/latex] or [latex]−3y–2\lt10[/latex]

Show Solution

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.

In the last example, the final answer included solutions whose intervals overlapped. This caused the answer to include all 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.

The following video contains an example of solving a compound inequality involving or and drawing the associated graph.

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]

Show Solution

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:

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]

Show Solution

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:

In the video below, you will see another example of how to solve an inequality in the form [latex]a<x<b[/latex]

To solve inequalities like [latex]a<x<b[/latex], use the addition and multiplication properties of inequality to solve the inequality for 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

In the example below, there is no solution to the compound inequality because there is no overlap between the inequalities.

Example

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

Show Solution

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.