Compound Inequalities

A compound inequality is two inequalities joined together with the word “or” or the word “and”. When the two inequalities are joined with “or”, the solution set must satisfy either equation. For example, [latex]x\lt 2 \text{ or } x\ge 5[/latex] tells us that the [latex]x[/latex]-values must be either less than [latex]2[/latex] or greater than or equal to [latex]5[/latex]. On the other hand, [latex]x\ge 2\text{ and }x\lt 5[/latex] tells us that the [latex]x[/latex]-values must be greater than or equal to [latex]2[/latex] and also less than [latex]5[/latex]. When “or” is used, the solution set is a union of the solution sets of both inequalities. When “and” is used, the solution set is an intersection of the solution sets of both inequalities.

Unions

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.

The following example shows 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. We will use the same properties to solve compound inequalities that we used to solve single inequalities.

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.

Intersections

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. 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.

Notice that the two inequalities [latex]x\gt -1[/latex] and [latex]x\leq 1[/latex] can be combined and written as [latex]-1\lt x\leq 1[/latex]. This is best seen on the graph where the solution set lies between [latex]-1[/latex] and [latex]1[/latex]. In other words, [latex]x[/latex] is trapped between [latex]-1[/latex] and [latex]1[/latex]. All intersection inequalities [latex]x\gt a[/latex] and [latex]x\lt b[/latex] can be written in the form [latex]a\lt x\lt b[/latex] provided [latex]a\lt b[/latex]. If [latex]a\gt b[/latex] there is no graphical overlap which means there is no solution.

Intersections in the form [latex]a

A compound inequality that is an intersection can be written in the form [latex]a\lt x\lt b[/latex] rather than [latex]x\gt a \text{ and }x\lt b[/latex] provided [latex]a\lt b[/latex]. If [latex]a\gt b[/latex] there is no solution. Rather than splitting a compound inequality in the form of  [latex]a and [latex]x\lt b[/latex], we can more quickly 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]

 

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

Set-builder notation: [latex]\{\;x\;\large |\normalsize\; 0\lt{x}\le 2,\;x\in\mathbb{R}\}[/latex]

Intervalotation n: [latex]x\in\left(0,2\right][/latex]

Graph:

 

The video below shows another example of how to solve an inequality in the form [latex]a

 

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] 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.

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]aand. 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.