Compound Inequalities

Compound Inequalities 

A compound inequality is a sentence with two inequality statements joined either by the word “or” or by the word “and.” “And” indicates that both statements of the compound sentence are true at the same time. It is the overlap or intersection of the solution sets for the individual statements. “Or” indicates that, as long as either statement is true, the entire compound sentence is true. It is the combination or union of the solution sets for the individual statement.

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What is compound inequality?

A compound inequality that uses the word “and” is known as a conjunction. Although “and” and “or” are parts of speech known as conjunctions, the mathematical conjunction has a different meaning from the grammatical one. To prove the point, the conjunction (part of speech) “or”—when used in a compound inequality—forms what is known as a disjunction. “con” means “with another,” and “dis” means “one OR the other.”

Example 1

Solve for x: 3 x + 2 < 14 and 2 x – 5 > –11.

Solve each inequality separately. Since the joining word is “and,” this indicates that the overlap or intersection is the desired result.

x < 4 indicates all the numbers to the left of 4, and x > –3 indicates all the numbers to the right of –3. The intersection of these two graphs is all the numbers between –3 and 4. The solution set is

{x| x > –3 and x < 4}

Example 2

Solve for x: 2 x + 7 < –11 or –3 x – 2 < 13.

Solve each inequality separately. Since the joining word is “or,” combine the answers; that is, find the union of the solution sets of each inequality sentence.

Remember, as in the last step on the right, to switch the inequality when multiplying by a negative.

x < –9 indicates all the numbers to the left of –9, and x > –5 indicates all the numbers to the right of –5. The solution set is written as

{x| x < –9 or x > –5}

Example 3

Solve for x: 3 x – 2 > –8 or 2 x + 1 < 9.

x > –2 indicates all the numbers to the right of –2, and x < 4 indicates all the numbers to the left of 4. The union of these graphs is the entire number line. That is, the solution set is all real numbers. The graph of the solution set is the entire number line

Arrow heads indicate infinity.

Example with "OR" 

Solve for x: 2 x + 7 < –11 or –3 x – 2 < 13.

Solve each inequality separately. Since the joining word is “or,” combine the answers; that is, find the union of the solution sets of each inequality sentence.

Remember, as in the last step on the right, to switch the inequality when multiplying by a negative.

x < –9 indicates all the numbers to the left of –9, and x > –5 indicates all the numbers to the right of –5. The solution set is written as

{x| x < –9 or x > –5}

Example with "AND"

Solve for x: 3 x + 2 < 14 and 2 x – 5 > –11.

Solve each inequality separately. Since the joining word is “and,” this indicates that the overlap or intersection is the desired result.

x < 4 indicates all the numbers to the left of 4, and x > –3 indicates all the numbers to the right of –3. The intersection of these two graphs is all the numbers between –3 and 4. The solution set is

{x| x > –3 and x < 4}

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Solving compound inequalities

To solve a compound inequality, first separate it into two inequalities. Determine whether the answer should be a union of sets ("or") or an intersection of sets ("and"). Then, solve both inequalities and graph.

A compound inequality contains at least two inequalities that are separated by either "and" or "or".

The graph of a compound inequality with an "and" represents the intersection of the graph of the inequalities. A number is a solution to the compound inequality if the number is a solution to both inequalities. It can either be written as x > -1 and x < 2 or as -1 < x < 2.

The graph of a compound inequality with an "or" represents the union of the graphs of the inequalities. A number is a solution to the compound inequality if the number is a solution to at least one of the inequalities. It is written as x < -1 or x > 2

Solve 

6x−3<9 and 2x+9≥3. Graph the solution and write the solution in interval notation.

Step 1.

Solve each inequality.

6x−3<9   2x+9≥3

 6x<12               2x≥−6

 x<2      and              x≥−3

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.

Example with "OR"

Solve for x: 2 x + 7 < –11 or –3 x – 2 < 13.

Solve each inequality separately. Since the joining word is “or,” combine the answers; that is, find the union of the solution sets of each inequality sentence.

Remember, as in the last step on the right, to switch the inequality when multiplying by a negative.

x < –9 indicates all the numbers to the left of –9, and x > –5 indicates all the numbers to the right of –5. The solution set is written as

{x| x < –9 or x > –5}

Example with "AND"

Solve for x: 3 x + 2 < 14 and 2 x – 5 > –11.

Solve each inequality separately. Since the joining word is “and,” this indicates that the overlap or intersection is the desired result.

x < 4 indicates all the numbers to the left of 4, and x > –3 indicates all the numbers to the right of –3. The intersection of these two graphs is all the numbers between –3 and 4. The solution set is

{x| x > –3 and x < 4}

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