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 statements. 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. Just remember “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.

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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}

Another way this solution set could be expressed is

{ x|–3 < x < 4}

When a compound inequality is written without the expressed word “and” or “or,” it is understood to automatically be the word “and.” Reading { x|–3 < x < 4} from the “ x” position, you say (reading to the left), “ x is greater than –3 and (reading to the right) x is less than 4.” The graph of the solution set is shown in Figure 1.

Figure 1. x is greater than –3 and less than 4.

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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.

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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}

The graph of this solution set is shown in Figure 2.

Figure 2. x is less than –9 or is greater than –5.

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Example 3

Solve for x: –12 ≤ 2 x + 6 ≤ 8.

Since this compound inequality has no connecting word written, it is understood to be “and.” It is translated into the following compound sentence.

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–9 ≤ x indicates all the numbers to the right of and including –9, and x ≤ 1 indicates all the numbers to the left of and including 1. The intersection of these graphs is the numbers between –9 and 1, including –9 and 1. The solution set can be written as

{ x| x ≥ –9 and x ≤ 1} or { x|–9 ≤ x ≤ 1}

The graph of the solution set is shown in Figure 3.

Figure 3. Dots indicate inclusion of the points.

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Example 4

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

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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 (see Figure 4).

Figure 4. Arrow heads indicate infinity.

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Example 5

Solve for x: 4 x – 2 < 10 and 3 x + 1 > 22.

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x < 3 indicates all the numbers to the left of 3, and x > 7 indicates all the numbers to the right of 7. The intersection of these graphs contains no numbers. That is, the solution set is the empty set, equation. A way to graph the empty set is to draw a number line but not to darken in any part of it. The graph of the empty set is shown in Figure 5.

Figure 5. The empty set.

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