Solve each inequality. Give the solution set in interval notation.
$\mid 5-3x\mid \le 7$

Verified step by step guidance

1. Start by understanding that the inequality involves an absolute value: \(|5 - 3x| \leq 7\). Recall that \(|A| \leq B\) means \(-B \leq A \leq B\) for any real numbers \(A\) and \(B \geq 0\).
2. Apply this property to the inequality: write it as a compound inequality: \(-7 \leq 5 - 3x \leq 7\).
3. Next, solve the left part of the compound inequality: \(-7 \leq 5 - 3x\). Subtract 5 from both sides to isolate the term with \(x\): \(-7 - 5 \leq -3x\), which simplifies to \(-12 \leq -3x\).
4. Divide both sides of the inequality by \(-3\) to solve for \(x\). Remember that dividing by a negative number reverses the inequality sign: \(\frac{-12}{-3} \geq x\), which simplifies to \(4 \geq x\) or \(x \leq 4\).
5. Now solve the right part of the compound inequality: \(5 - 3x \leq 7\). Subtract 5 from both sides: \(-3x \leq 2\). Divide both sides by \(-3\), reversing the inequality sign: \(x \geq -\frac{2}{3}\). Combine both parts to write the solution set in interval notation.

Key Concepts

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, rewrite it as -B ≤ A ≤ B, converting it into a compound inequality that can be solved using standard algebraic methods.

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. Solutions are often expressed as intervals or inequalities.

Interval Notation

Interval notation is a concise way to represent sets of numbers that satisfy inequalities. It uses parentheses () for values not included and brackets [] for values included. For example, [a, b] represents all numbers between a and b, including both endpoints.