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Mathematics LibreTexts

19.2: Solve Compound Inequalities

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Learning Objectives

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

  • Solve compound inequalities with “and”
  • Solve compound inequalities with “or”
  • Solve applications with compound inequalities

Before you get started, take this readiness quiz.

  1. Simplify: 25(x+10).
    If you missed this problem, review [link].
  2. Simplify: (x4).
    If you missed this problem, review [link].

Solve Compound Inequalities with “and”

Now that we know how to solve linear inequalities, the next step is to look at compound inequalities. A compound inequality is made up of two inequalities connected by the word “and” or the word “or.” For example, the following are compound inequalities.

x+3>4and4x532(y+1)<0ory52

COMPOUND INEQUALITY

A compound inequality is made up of two inequalities connected by the word “and” or the word “or.”

To solve a compound inequality means to find all values of the variable that make the compound inequality a true statement. We solve compound inequalities using the same techniques we used to solve linear inequalities. We solve each inequality separately and then consider the two solutions.

To solve a compound inequality with the word “and,” we look for all numbers that make both inequalities true. To solve a compound inequality with the word “or,” we look for all numbers that make either inequality true.

Let’s start with the compound inequalities with “and.” Our solution will be the numbers that are solutions to both inequalities known as the intersection of the two inequalities. Consider the intersection of two streets—the part where the streets overlap—belongs to both streets.

The figure is an illustration of two streets with their intersection shaded

To find the solution of an "and" compound inequality, we look at the graphs of each inequality and then find the numbers that belong to both graphs—where the graphs overlap.

For the compound inequality x>3 and x2, we graph each inequality. We then look for where the graphs “overlap”. The numbers that are shaded on both graphs, will be shaded on the graph of the solution of the compound inequality. See Figure 19.2.1.

The figure shows the graph of x is greater than negative 3 with a left parenthesis at negative 3 and shading to its right, the graph of x is less than or equal to 2 with a bracket at 2 and shading to its left, and the graph of x is greater than negative 3 and x is less than or equal to 2 with a left parenthesis at negative 3 and a right parenthesis at 2 and shading between negative 3 and 2. Negative 3 and 2 are marked by lines on each number line.
Figure 19.2.1

We can see that the numbers between 3 and 2 are shaded on both of the first two graphs. They will then be shaded on the solution graph.

The number 3 is not shaded on the first graph and so since it is not shaded on both graphs, it is not included on the solution graph.

The number two is shaded on both the first and second graphs. Therefore, it is be shaded on the solution graph.

This is how we will show our solution in the next examples.

Example 19.2.1

Solve 6x3<9 and 2x+73. Graph the solution and write the solution in interval notation.

Solution

  6x3<9 and 2x+93
Step 1. Solve each
inequality.
6x3<9   2x+93
  6x<12   2x6
  x<2 and x3
Step 2. Graph each solution. Then graph the numbers that make both inequalities true. The final graph will show all the numbers that make both inequalities true—the numbers shaded on both of the first two graphs. .
Step 3. Write the solution in interval notation. [3,2)
All the numbers that make both inequalities true are the solution to the compound inequality.
Try It 19.2.2

Solve the compound inequality. Graph the solution and write the solution in interval notation: 4x7<9 and 5x+83.

Answer

The solution is negative 1 is less than or equal to x which is less than 4. On a number line it is shown with a closed circle at negative 1 and an open circle at 4 with shading in between the closed and open circles. Its interval notation is negative 1 to 4 within a bracket and a parenthesis.

Try It 19.2.3

Solve the compound inequality. Graph the solution and write the solution in interval notation: 3x4<5 and 4x+91.

Answer

The solution is negative 2 is less than or equal to x which is less than 3. On a number line it is shown with a closed circle at negative 2 and an open circle at 3 with shading in between the closed and open circles. Its interval notation is negative 2 to 3 within a bracket and a parenthesis.

SOLVE A COMPOUND INEQUALITY WITH “AND.”
  1. Solve each inequality.
  2. Graph each solution. Then graph the numbers that make both inequalities true.
    This graph shows the solution to the compound inequality.
  3. Write the solution in interval notation.
Example 19.2.4

Solve 3(2x+5)18 and 2(x7)<6. Graph the solution and write the solution in interval notation.

Solution

  3(2x+5)18 and 2(x7)<6
Solve each
inequality.
6x+1518   2x14<6
  6x3   2x<8
  x12 and x<4
Graph each
solution.
.
Graph the numbers
that make both
inequalities true.
.
Write the solution
in interval notation.
(,12]
Try It 19.2.5

Solve the compound inequality. Graph the solution and write the solution in interval notation: 2(3x+1)20 and 4(x1)<2.

Answer

The solution is x is less than three-halves. On a number line it is shown with an open circle at three-halves with shading to its left. Its interval notation is negative infinity to three-halves within a parentheses.

Try It 19.2.6

Solve the compound inequality. Graph the solution and write the solution in interval notation: 5(3x1)10 and 4(x+3)<8.

Answer

The solution is x is less than negative 1. On a number line it is shown with an open circle at 1 with shading to its left. Its interval notation is negative infinity to negative 1 within parentheses.

Example 19.2.7

Solve 13x42 and 2(x3)4. Graph the solution and write the solution in interval notation.

Solution

  13x42 and 2(x3)4
Solve each inequality. 13x42   2x+64
  13x2   2x2
  x6 and x1
Graph each solution. .
Graph the numbers that
make both inequalities
true.
.
  This is a contradiction. There are no numbers that make both inequalities true, so there is no solution. 
Try It 19.2.8

Solve the compound inequality. Graph the solution and write the solution in interval notation: 14x31 and 3(x2)2.

Answer

The inequality is a contradiction. So, there is no solution. As a result, there is no graph of the number line or interval notation.

Try It 19.2.9

Solve the compound inequality. Graph the solution and write the solution in interval notation: 15x53 and 4(x1)2.

Answer

The inequality is a contradiction. So, there is no solution. As a result, there is no graph or the number line or interval notation.

Sometimes we have a compound inequality that can be written more concisely. For example, a<x and x<b can be written simply as a<x<b and then we call it a double inequality. The two forms are equivalent.

DOUBLE INEQUALITY

A double inequality is a compound inequality such as a<x<b. It is equivalent to a<x and x<b.

Other forms:a<x<bis equivalent to a<xandx<baxbis equivalent to axandxba>x>bis equivalent to a>xandx>baxbis equivalent to axandxb

To solve a double inequality we perform the same operation on all three “parts” of the double inequality with the goal of isolating the variable in the center.

Example 19.2.10

Solve 43x7<8. Graph the solution and write the solution in interval notation.

Solution

  43x7<8
Add 7 to all three parts. 4+73x7+7<8+7
Simplify. 33x<15
Divide each part by three. 333x3<153
Simplify. 1x<5
Graph the solution. .
Write the solution in interval notation. [1,5)

When written as a double inequality, 1x<5, it is easy to see that the solutions are the numbers caught between one and five, including one, but not five. We can then graph the solution immediately as we did above.

Another way to graph the solution of 1x<5 is to graph both the solution of x1 and the solution of x<5. We would then find the numbers that make both inequalities true as we did in previous examples.

Try It 19.2.11

Solve the compound inequality. Graph the solution and write the solution in interval notation: 54x1<7.

Answer

The solution is negative 1 is less than or equal to x which is less than 2. Its graph has a closed circle at negative 1 and an open circle at 2 with shading between the closed and open circles. Its interval notation is negative 1 to 2 within a bracket and a parenthesis.

Try It 19.2.12

Solve the compound inequality. Graph the solution and write the solution in interval notation: 3<2x51.

Answer

The solution is 1 is less than x which is less than or equal to 3. Its graph has an open circle at 1 and a closed circle at 3 with shading between the closed and open circles. Its interval notation is negative 1 to 3 within a parenthesis and a bracket.

Solve Compound Inequalities with “or”

To solve a compound inequality with “or”, we start out just as we did with the compound inequalities with “and”—we solve the two inequalities. Then we find all the numbers that make either inequality true.

Just as the United States is the union of all of the 50 states, the solution will be the union of all the numbers that make either inequality true. To find the solution of the compound inequality, we look at the graphs of each inequality, find the numbers that belong to either graph and put all those numbers together.

To write the solution in interval notation, we will often use the union symbol, , to show the union of the solutions shown in the graphs.

SOLVE A COMPOUND INEQUALITY WITH “OR.”
  1. Solve each inequality.
  2. Graph each solution. Then graph the numbers that make either inequality true.
  3. Write the solution in interval notation.
Example 19.2.13

Solve 53x1 or 8+2x5. Graph the solution and write the solution in interval notation.

Solution

  53x1 or 8+2x5
Solve each inequality. 53x1   8+2x5
  3x6   2x3
  x2 or x32
Graph each solution. .
Graph numbers that
make either inequality
true.
.
  (,32][2,)
Try It 19.2.14

Solve the compound inequality. Graph the solution and write the solution in interval notation: 12x3 or 7+3x4.

Answer

The solution is x is greater than or equal to 2 or x is less than or equal to 1. The graph of the solutions on a number line has a closed circle at negative 1 and shading to the left and a closed circle at 2 with shading to the right. The interval notation is the union of negative infinity to negative 1 within a parenthesis and a bracket and 2 and infinity within a bracket and a parenthesis.

Try It 19.2.15

Solve the compound inequality. Graph the solution and write the solution in interval notation: 25x3 or 5+2x3.

Answer

The solution is x is greater than or equal to 1 or x is less than or equal to negative 1. The graph of the solutions on a number line has a closed circle at negative 1 and shading to the left and a closed circle at 1 with shading to the right. The interval notation is the union of negative infinity to negative 1 within a parenthesis and a bracket and 1 and infinity within a bracket and a parenthesis.

Example 19.2.16

Solve 23x43 or 14(x+8)1. Graph the solution and write the solution in interval notation.

Solution

  23x43 or 14(x+8)1
Solve each
inequality.
3(23x4)3(3)   414(x+8)4(1)
  2x129   x+84
  2x21   x12
  x212    
  x212 or x12
Graph each
solution.
.
Graph numbers
that make either
inequality true.
.
  The solution covers all real numbers.
  (,)
Try It 19.2.17

Solve the compound inequality. Graph the solution and write the solution in interval notation: 35x71 or 13(x+6)2.

Answer

The solution is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.

Try It 19.2.18

Solve the compound inequality. Graph the solution and write the solution in interval notation: 34x33 or 25(x+10)0.

Answer

The solution is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.

Solve Applications with Compound Inequalities

Situations in the real world also involve compound inequalities. We will use the same problem solving strategy that we used to solve linear equation and inequality applications.

Recall the problem solving strategies are to first read the problem and make sure all the words are understood. Then, identify what we are looking for and assign a variable to represent it. Next, restate the problem in one sentence to make it easy to translate into a compound inequality. Last, we will solve the compound inequality.

Example 19.2.19

Due to the drought in California, many communities have tiered water rates. There are different rates for Conservation Usage, Normal Usage and Excessive Usage. The usage is measured in the number of hundred cubic feet (hcf) the property owner uses.

During the summer, a property owner will pay $24.72 plus $1.54 per hcf for Normal Usage. The bill for Normal Usage would be between or equal to $57.06 and $171.02. How many hcf can the owner use if he wants his usage to stay in the normal range?

Solution

Identify what we are looking for. The number of hcf he can use and stay in the “normal usage” billing range.
Name what we are looking for. Let x= the number of hcf he can use.
Translate to an inequality. Bill is $24.72 plus $1.54 times the number of hcf he uses or 24.72+1.54x.
 

His bill will be between or equal to $57.06 and $171.02.

57.0624.72+1.54x171.02

Solve the inequality.

57.0624.72+1.54x171.02

57.0624.7224.7224.72+1.54x171.0224.72

32.341.54x146.3

32.341.541.54x1.54146.31.54

21x95

Answer the question. The property owner can use 2195 hcf and still fall within the “normal usage” billing range.
Try It 19.2.20

Due to the drought in California, many communities now have tiered water rates. There are different rates for Conservation Usage, Normal Usage and Excessive Usage. The usage is measured in the number of hundred cubic feet (hcf) the property owner uses.

During the summer, a property owner will pay $24.72 plus $1.32 per hcf for Conservation Usage. The bill for Conservation Usage would be between or equal to $31.32 and $51.12. How many hcf can the owner use if she wants her usage to stay in the conservation range?

Answer

The homeowner can use 520 hcf and still fall within the “conservation usage” billing range.

Try It 19.2.21

Due to the drought in California, many communities have tiered water rates. There are different rates for Conservation Usage, Normal Usage and Excessive Usage. The usage is measured in the number of hundred cubic feet (hcf) the property owner uses.

During the winter, a property owner will pay $24.72 plus $1.54 per hcf for Normal Usage. The bill for Normal Usage would be between or equal to $49.36 and $86.32. How many hcf will he be allowed to use if he wants his usage to stay in the normal range?

Answer

The homeowner can use 1640  hcf and still fall within the “normal usage” billing range.

Access this online resource for additional instruction and practice with solving compound inequalities.

  • Compound inequalities

Key Concepts

  • How to solve a compound inequality with “and”
    1. Solve each inequality.
    2. Graph each solution. Then graph the numbers that make both inequalities true. This graph shows the solution to the compound inequality.
    3. Write the solution in interval notation.
  • Double Inequality
    • A double inequality is a compound inequality such as a<x<b. It is equivalent to a<x and x<b.

      Other forms: a<x<bis equivalent toa<xandx<baxbis equivalent toaxandxba>x>bis equivalent toa>xandx>baxbis equivalent toaxandxb
  • How to solve a compound inequality with “or”
    1. Solve each inequality.
    2. Graph each solution. Then graph the numbers that make either inequality true.
    3. Write the solution in interval notation.

Glossary

compound inequality
A compound inequality is made up of two inequalities connected by the word “and” or the word “or.”

19.2: Solve Compound Inequalities is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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