Solving Compound Inequalities

Learning Outcomes

By the end of this section, you will be able to:

  • Solve compound inequalities involving “or,” and express solutions both graphically and with interval notation
  • Solve compound inequalities involving “and,” and express solutions both graphically and with interval notation

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]

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]

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.

Example

Solve for [latex]z[/latex].

[latex]5z–3\gt−18[/latex] or [latex]−2z–1\gt15[/latex]

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

Try It

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]

EXample

Solve for x:

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

Try It

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]

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 Number line. Open blue circle on negative 1 and blue arrow through all numbers greater than negative 1. The blue arrow represents x is greater than negative 1. Closed red circle on 1 and red arrow through all numbers less than 1. Red arrow written x is less than or equal to 1.

Number line. Open blue circle on negative 1. Closed red circle on 1. Overlapping red and blue lines between negative 1 and 1 that represents negative 1 is less than x is less than or equal to 1.

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  Number line. Blue open circle on negative 3 and blue arrow through all numbers greater than negative 3. Blue arrow represents x is greater than negative three. Closed red circle on 4 and red arrow through all numbers greater than 4. The red arrow respresents x is greater than or equal to 4.

Number line. Closed circle on 4 and arrow through all numbers greater than 4. The arrow represents x is greater than or equal to 4.

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 Number line. Open red circle on negative 3 and red arrow through all numbers less than negative 3. Red arrow represents x is less than negative 3. Open blue circle on 3 and blue arrow through all numbers greater than 3. Blue arrow represents x is greater than 3.

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]