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Solve Compound Inequalities

Author: Sophia
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what's covered
This section will cover the intersection and union of intervals. Compound inequalities will be introduced along with how to solve a compound inequality. We will see real-world applications of solving compound inequalities. You'll also explore the connection between linear equations and building your results driven skill. Specifically, this lesson will cover:

Table of Contents

before you start
Sometimes we need to join two or more inequalities together, we can do this by using what we call compound inequalities. Compound inequalities are at least two inequality statements joined together by the word “or” or by the word “and". Compound inequalities can be formed from the union of inequalities or the intersection of inequalities.

1. Union of Inequalities ("OR" Inequalities)

Let’s first look at a union of inequalities. This will be the case when you have more than one highlighted range on your number line. For a union of inequalities, we want a true statement from either one inequality OR the other inequality OR both. We can sometimes call these “or” inequalities. This will be similar to interval notation when there is more than one highlighted range; we use the ∪ (union) symbol between two sets of interval notation. For example, open parentheses short dash infinity comma space 3 close square brackets union open parentheses 4 comma space infinity close parentheses. Again, this means that x can either be in the range open parentheses short dash infinity comma space 3 close square brackets OR can be in the range open parentheses 4 comma space infinity close parentheses.

EXAMPLE

Solve each inequality for x. Write the solution in set notation, interval notation, and graph the solution.

2 x minus 7 greater than 3 space or space 4 minus x greater or equal than 6

Notice that we are given two different inequalities joined together by an “or” statement. We must first solve for x for each statement separately, then examine the ranges to determine a final solution.

2 x minus 7 greater than 3 The inequality 4 minus x greater or equal than 6 The inequality
2 x minus 7 plus 7 greater than 3 plus 7 Add 7 to both sides of the inequality. 4 minus 4 minus x greater or equal than 6 minus 4 Subtract 4 from both sides of the inequality.
2 x greater than 10 Isolate the variable by addressing the coefficient attached to the variable. negative x greater or equal than 2 Isolate the variable by addressing the coefficient attached to the variable.
fraction numerator 2 x over denominator 2 end fraction greater than 10 over 2 Divide both sides by 2. fraction numerator negative x over denominator negative 1 end fraction greater than fraction numerator 2 over denominator negative 1 end fraction Divide both sides by -1.
x greater than 5 x is greater than 5. x less or equal than negative 2 Flip the sign of the inequality. x is less than or equal to -2.

Our two inequalities result in two intervals that do not overlap. Because we are given the union of two inequalities, x can be in either interval.

Compound Inequality
2 x minus 7 greater than 3 space or space 4 minus x greater or equal than 6
Solutions
Set notation:
{x | x ≤ -2 or x > 5} 		Interval notation:
left parenthesis negative infinity comma negative 2 right square bracket union left parenthesis 5 comma infinity right parenthesis
Graph on the number line
Number line with two ranges highlighted. -2 has a closed circle and all values below are highlighted. 5 has an open circle and all values above are highlighted

EXAMPLE

Solve each inequality for x. Write the solution in set notation, interval notation, and graph the solution.

3 open parentheses x minus 2 close parentheses less than 9 space or space short dash x minus 2 greater or equal than short dash 6

3 left parenthesis x minus 2 right parenthesis less than 9 The inequality short dash x minus 2 greater or equal than short dash 6 The inequality
3 x minus 6 less than 9 Use the distributive property. short dash x minus 2 plus 2 greater or equal than short dash 6 plus 2 Add 2 to both sides of the inequality.
3 x minus 6 plus 6 less than 9 plus 6 Add 6 to both sides of the equation. short dash x greater or equal than short dash 4 Isolate the variable by addressing the coefficient attached to the variable.
3 x less than 15 Isolate the variable by addressing the coefficient attached to the variable. fraction numerator short dash x over denominator short dash 1 end fraction greater or equal than fraction numerator short dash 4 over denominator short dash 1 end fraction Divide both sides of the equation by -1. Remember to flip the inequality symbol.
fraction numerator 3 x over denominator 3 end fraction less than 15 over 3 Divide both sides of the inequality by 3. x less or equal than 4 x is less than or equal to 4.
x less than 5 x is less than 5.    

Notice that if we look at the two intervals on the number line, they overlap.

Number line with two highlighted ranges. 5 has an open circle and all values below are highlighted. 4 has a closed circle and all values below are highlighted

Because we are given the union of two inequalities, x can be in either interval or both. In this case, we would add the two intervals together and take the biggest interval or longest interval. This is x less than 5. Here we would only have one solution, x less than 5.

Compound Inequality
3 left parenthesis x minus 2 right parenthesis less than 9 space or space short dash x plus 2 greater or equal than 6
Solutions
Set notation:
{x | x < 5} 		Interval notation:
left parenthesis negative infinity comma 5 right parenthesis
Graph on the number line
Number line with one range highlighted. 5 has an open circle and all values below are highlighted.

term to know
Compound Inequality
Joining together two inequality statements with either "or" or "and".

2. Intersection of Inequalities

2a. "AND" Inequalities

The second type of compound inequality is the intersection of inequalities. The solution of this type of inequality is only where two or more inequalities overlap. When our solution is given in interval notation, it will be expressed in a manner very similar to single inequalities.

EXAMPLE

Solve each inequality for x. Write the solution in interval notation and graph the solution.

8 greater than 3 x minus 7 space and space 5 x greater or equal than 3 x plus 4

We start by solving each inequality separately and then examining the resulting intervals.

8 greater than 3 x minus 7 The inequality 5 x greater or equal than 3 x plus 4 The inequality
8 plus 7 greater than 3 x minus 7 plus 7 Add 7 to both sides of the inequality. 5 x minus 3 x greater or equal than 3 x minus 3 x plus 4 Subtract 3 x from both sides of the inequality.
15 greater than 3 x Isolate the variable by addressing the coefficient attached to the variable. 2 x greater or equal than 4 Isolate the variable by addressing the coefficient attached to the variable.
15 over 3 greater than fraction numerator 3 x over denominator 3 end fraction Divide both sides of the inequality by 3. fraction numerator 2 x over denominator 2 end fraction greater or equal than 4 over 2 Divide both sides of the inequality by 2.
5 greater than x x is less than 5. x greater or equal than 2 x is greater than or equal to 2.

Notice that if we look at the two intervals on the number line, they overlap.

Number line with two highlighted ranges. 5 has an open circle and all values below are highlighted. 2 has a closed circle and all values above are highlighted

Because we are asked to find the intersection of two inequalities, we only want solutions of x that are in both intervals. The x values that overlap in the two intervals are between 2 and 5. We would take the endpoints for each of the intervals. Here our solution for the intersection (or only the interval that overlaps) would be:

Compound Inequality
8 greater than 3 x minus 7 space and space 5 x greater or equal than 3 x plus 4
Solutions
Set notation:
{x | x < 5 and x ≥ 2} 		Interval notation:
left square bracket 2 comma 5 right parenthesis
Graph on the number line
Number line with one range highlighted between 2 and 5. 2 has a closed circle and 5 has an open circle

EXAMPLE

Solve each inequality for x. Write the solution in interval notation.

short dash 5 x plus 2 greater than 12 space and space 2 x greater than x plus 3

short dash 5 x plus 2 greater than 12 The inequality 2 x greater than x plus 3 The inequality
negative 5 x plus 2 minus 2 greater than 12 minus 2 Subtract 2 from both sides of the inequality. 2 x minus x greater than x minus x plus 3 Subtract x from both sides of the inequality.
negative 5 x greater than 10 Isolate the variable by addressing the coefficient attached to the variable. x greater than 3 x is greater than 3.
fraction numerator negative 5 x over denominator negative 5 end fraction greater than fraction numerator 10 over denominator negative 5 end fraction Divide both sides by -5 and flip the inequality sign.
x less than negative 2 x is less than -2.

We have two intervals, x less than negative 2 and x greater than 3. Let’s graph these intervals on the number line to find the overlap.

Number line with two highlighted portions, one starting at -2 with an open circle and all values below highlighted, and one starting at 3 with an open circle and all values above highlighted

Notice that the intervals do not overlap. There are no values of x that are contained in both intervals. So, there is no solution to this intersection of intervals.

Solutions:
Interval notation: No Solution

2b. Double Inequalities

Previously, when learning about set notation, if a value was between two values, we wrote the set, for example, as {x | x > 5 and x ≤ 8}. This indicates that x is a value between 5 and 8. We will learn to write this inequality as a single inequality called a compound inequality. In a compound inequality, when our variable (or expression containing the variable) is between two numbers, we can write it as a single math sentence with three parts, such as 5 less than x less than or equal to 8, to show x is between 5 and 8 (or equal to 8), or 5 less than x less or equal than 8.

EXAMPLE

Solve negative 6 less or equal than negative 4 x plus 2 less than 2. Write in interval notation and graph the solution.
Notice that there are 3 parts to the compound inequality.

table attributes columnalign left end attributes row cell negative 6 less or equal than negative 4 x plus 2 less than 2 end cell row cell ① space space space space space space space space space ② space space space space space space space space space space space ③ end cell end table

When solving a compound inequality for a variable, it is important to remember to stay balanced, meaning you will do the same operations to all three parts.

negative 6 less or equal than negative 4 x plus 2 less than 2 The compound inequality
negative 6 minus 2 less or equal than negative 4 x plus 2 minus 2 less than 2 minus 2 Subtract 2 from all 3 parts of the inequality.
negative 8 less or equal than negative 4 x less than 0 Isolate the variable by addressing the coefficient attached to the variable.
fraction numerator negative 8 over denominator negative 4 end fraction less or equal than fraction numerator negative 4 x over denominator negative 4 end fraction less than fraction numerator 0 over denominator negative 4 end fraction Divide all sides by -4.
2 greater or equal than x greater than 0 Flip all the inequality signs because you are dividing by a negative.
0 less than x less or equal than 2 Rearrange the inequality so the numbers are in increasing order.

The solution is 0 less than x less or equal than 2. This means that x can be any value that is greater than 0 and less than or equal to 2.

Compound Inequality
short dash 6 less or equal than short dash 4 x plus 2 less than 2
Solutions
Set notation:
{x | 0 < x ≤ 2} 		Interval notation:
left parenthesis 0 comma 2 right square bracket
Graph on the number line
Number line with one range highlighted between 0 and 2. 0 has an open circle and 2 has a closed circle

EXAMPLE

Most beagles have shoulder heights between 36 and 41 centimeters. The following compound inequality relates the estimated shoulder height (in centimeters) of a dog to the internal dimension of the skull D (in cubic centimeters):

36 less or equal than 1.02 D – 31.2 less or equal than 41

Find the compound inequality that represents the range for the skull size of beagles. Round to the nearest tenth.

36 less or equal than 1.02 D – 31.2 less or equal than 41 The compound inequality
67.2 less or equal than 1.02 D less or equal than 72.2 Isolate the variable D in the middle part by first addressing the constant. Add 31.2 to all 3 parts of the inequality.
65.9 less or equal than D less or equal than 70.8 Divide all 3 parts by 1.02. Solution rounded to the nearest tenth.

think about it
Chances are, your job won’t involve finding beagle skull dimensions, but it will probably involve compound inequalities at some point! For instance, you collaborate with a coworker on a presentation, and both of you have an average range of time that it takes to complete a slide. You could use inequalities to calculate an estimated time range for the project to be finished. Mastering inequalities will help you excel in your results driven skill, allowing you to work with urgency while ensuring that the work gets done on time.

summary
In this lesson, you learned about several types of compound inequalities, which have two or more inequalities. You learned that a union of inequalities occurs when you have more than one highlighted range on your number line. Sometimes referred to as “or” inequalities, for a union of inequalities, you want a true statement from either one inequality OR the other inequality OR both. You learned about the intersection of inequalities, noting that the solution of this type of inequality is only where two or more inequalities overlap. You also learned that in a compound inequality, when your variable (or expression containing the variable) is between two numbers, you can write it as a single math sentence with three parts. Remember that when solving a compound inequality for a variable, it is important to do the same operations to all three parts.

Best of luck in your learning!

Source: THIS TUTORIAL HAS BEEN ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE. ACCESS FOR FREE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License

Terms to Know
Compound Inequality

Joining together two inequality statements with either "or" or "and".

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