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# Compound Inequalities

If we join to simple inequalities with the connective "and" or the connective "or", we get a compound inequality. A compound inequality using the connective "and" is true if and only if both simple inequalities are true.

Example 1

Compound inequalities using the connective "and"

Determine whether each compound inequality is true.

a) 3 > 2 and 3 < 5

b) 6 > 2 and 6 < 5

Solution

a) The compound inequality is true because 3 > 2 is true and 3 < 5 is true.

b) The compound inequality is false because 6 < 5 is false.

A compound inequality using the connective "or" is true if one or the other or both of the simple inequalities are true. It is false only if both simple inequalities are false.

Example 2

Compound inequalities using the connective "or"

Determine whether each compound inequality is true.

a) 2 < 3 or 2 > 7

b) 4 < 3 or 4 ≥ 7

Solution

a) The compound inequality is true because 2 < 3 is true.

b) The compound inequality is false because both 4 < 3 and 4 ≥ 7 are false.

If a compound inequality involves a variable, then we are interested in the solution set to the inequality. The solution set to an "and" inequality consists of all numbers that satisfy both simple inequalities, whereas the solution set to an "or" inequality consists of all numbers that satisfy at least one of the simple inequalities.

Example 3

Solutions of compound inequalities

Determine whether 5 satisfies each compound inequality.

a) x < 6 and x < 9

b) 2x - 9 ≤ 5 or -4x ≥ -12

Solution

a) Because 5 < 6 and 5 < 9 are both true, 5 satisfies the compound inequality.

b) Because 2 Â· 5 - 9 ≤ 5 is true, it does not matter that -4 Â· 5 ≥ -12 is false. So 5 satisfies the compound inequality.