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

Verified step by step guidance

1. Start by considering the definition of absolute value inequalities. The inequality \(|5 - 3x| > 7\) can be split into two separate inequalities: \(5 - 3x > 7\) and \(5 - 3x < -7\).
2. Solve the first inequality \(5 - 3x > 7\). Subtract 5 from both sides to isolate the term with \(x\): \(-3x > 2\).
3. Divide both sides of the inequality \(-3x > 2\) by \(-3\). Remember to reverse the inequality sign when dividing by a negative number: \(x < -\frac{2}{3}\).
4. Solve the second inequality \(5 - 3x < -7\). Subtract 5 from both sides: \(-3x < -12\).
5. Divide both sides of the inequality \(-3x < -12\) by \(-3\), again reversing the inequality sign: \(x > 4\).

Key Concepts

Absolute Value Inequalities

Absolute value inequalities involve expressions that measure the distance of a number from zero on the number line. The inequality |A| > B indicates that A is either greater than B or less than -B. Understanding how to break down these inequalities into two separate cases is crucial for finding the solution set.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It uses parentheses and brackets to indicate whether endpoints are included (closed interval) or excluded (open interval). For example, (a, b) means all numbers between a and b, not including a and b, while [a, b] includes both endpoints.

Solving Linear Inequalities

Solving linear inequalities involves finding the values of a variable that satisfy the inequality. This process often includes isolating the variable on one side of the inequality sign and may require reversing the inequality when multiplying or dividing by a negative number. The solution is typically expressed in interval notation.