Problem 137 - College Algebra

Question

What is a compound inequality and how is it solved?

Step-by-Step Solution

Answer

A compound inequality is two inequalities combined into one statement by either 'and' or 'or'. If the inequalities are connected by 'and', the solution must satisfy both inequalities. If by 'or', the solution can satisfy either one. To solve, each inequality is solved individually. For 'and', the solution is the intersection of the two solutions, and for 'or', the solution could be in one or both ranges, depending where they are.

Step 1: Understanding Compound Inequalities

A compound inequality is an equation with two inequalities that are joined together with either 'and' or 'or'. If the inequalities are connected by 'and', then the solution must satisfy both inequalities. If they are connected by 'or', then the solution can satisfy either one of the inequalities. For example, a compound inequality might be \(3x + 2 > 1 \text{ and } 2x - 1 < 3\).

Step 2: Solving 'And' Compound Inequalities

For compound inequalities joined by 'and', both inequalities must be solved simultaneously. Using the example compound inequality \(3x + 2 > 1 \text{ and } 2x - 1 < 3\), solve each inequality as if they were separate. First, for \(3x + 2 > 1\) we subtract 2 from both sides to get \(3x > -1\), then divide through by 3 to get \(x > -1/3\). For the inequality \(2x - 1 < 3\), we follow a similar process to get \(x < 2\). Since it's an 'and' compound inequality, the overall solution is the overlap of these two individual solutions, that is \(-1/3 < x < 2\).

Step 3: Solving 'Or' Compound Inequalities

For compound inequalities joined by 'or', either inequality can be true. If we had an inequality like \(3x + 2 > 1 \text{ or } 2x - 1 < 3\), we could solve each inequality and any solution of either inequality would be a valid solution. So in this case any \(x\) satisfying \(x > -1/3\) or \(x < 2\) would be a solution. Typically, an 'or' compound inequality might include values over a wider range or even multiple separate ranges.