Solve each inequality. Give the solution set in interval notation. | 5 - 3x | > 7

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Start by understanding that the inequality involves an absolute value: \(|5 - 3x| > 7\). Recall that \(|A| > B\) means \(A > B\) or \(A < -B\) when \(B > 0\).

Set up two separate inequalities based on the definition of absolute value: 1) \(5 - 3x > 7\) 2) \(5 - 3x < -7\)

Solve the first inequality \(5 - 3x > 7\): Subtract 5 from both sides to isolate the term with \(x\): \(-3x > 7 - 5\) Simplify the right side: \(-3x > 2\) Divide both sides by \(-3\), remembering to reverse the inequality sign because you are dividing by a negative number: \(x < \frac{2}{-3}\)

Solve the second inequality \(5 - 3x < -7\): Subtract 5 from both sides: \(-3x < -7 - 5\) Simplify the right side: \(-3x < -12\) Divide both sides by \(-3\), reversing the inequality sign: \(x > \frac{-12}{-3}\)

Combine the two solution sets from the inequalities to express the solution in interval notation. The solution will be all \(x\) values less than \(\frac{2}{-3}\) or greater than \(\frac{-12}{-3}\).

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주요 개념

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Absolute Value Inequalities

Absolute value inequalities involve expressions where the absolute value of a variable or expression is compared to a number. To solve |A| > B, where B > 0, split the inequality into two cases: A > B or A < -B. This approach helps find all values of the variable that satisfy the inequality.

Solving Linear Inequalities

Solving linear inequalities requires isolating the variable on one side while maintaining the inequality's direction. When multiplying or dividing by a negative number, the inequality sign must be reversed. This process yields the range of values that satisfy the inequality.

Interval Notation

Interval notation is a concise way to represent solution sets of inequalities using intervals. Parentheses () denote values not included (open intervals), while brackets [] denote included values (closed intervals). It clearly shows the range of solutions on the number line.