Understanding and Solving Compound Inequalities: A Comprehensive Guide

compound inequality

A compound inequality is an equation that contains two or more inequalities joined together by the words “and” or “or

A compound inequality is an equation that contains two or more inequalities joined together by the words “and” or “or.” The purpose of a compound inequality is to represent a range of values that satisfy both or either of the inequalities simultaneously.

There are two types of compound inequalities: “and” compound inequalities and “or” compound inequalities.

1. “And” Compound Inequalities:
In the case of an “and” compound inequality, both inequalities must be true at the same time for a solution to exist. The symbol used to represent “and” is an upside-down “U” (∩). Let’s consider an example to illustrate this type of compound inequality:

-2 < x + 4 ≤ 6 To solve this compound inequality, we need to solve both inequalities separately. First inequality, -2 < x + 4: To isolate x, we subtract 4 from both sides of the inequality: -2 - 4 < x + 4 - 4 -6 < x Second inequality, x + 4 ≤ 6: To isolate x, we subtract 4 from both sides of the inequality: x + 4 - 4 ≤ 6 - 4 x ≤ 2 Combining both inequalities, we have -6 < x ≤ 2. This means that x can take any value between -6 and 2, including -6 but not including 2. 2. "Or" Compound Inequalities: In the case of an "or" compound inequality, either of the inequalities can be true for a solution to exist. The symbol used to represent "or" is a union symbol (∪). Let's consider an example: 4x + 1 < 9 or 3x - 2 ≥ 5 To solve this compound inequality, we need to solve both inequalities separately. First inequality, 4x + 1 < 9: To isolate x, we subtract 1 from both sides of the inequality: 4x + 1 - 1 < 9 - 1 4x < 8 x < 2 Second inequality, 3x - 2 ≥ 5: To isolate x, we add 2 to both sides of the inequality: 3x - 2 + 2 ≥ 5 + 2 3x ≥ 7 x ≥ 7/3 Combining both inequalities using the "or" symbol, we have x < 2 or x ≥ 7/3. This means that x can take any value less than 2 or greater than or equal to 7/3. These are the basic steps to solve compound inequalities. Remember to address each inequality separately and combine the solutions according to the compound inequality given.

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