Solve each inequality. Give the solution set in interval notation. | 7 - 3x | ≤ 4

Verified step by step guidance

1. Recognize that the inequality involves an absolute value: \(|7 - 3x| \leq 4\). Recall that \(|A| \leq B\) means \(-B \leq A \leq B\) for any real numbers \(A\) and \(B \geq 0\).
2. Rewrite the inequality without the absolute value by setting up a compound inequality: \(-4 \leq 7 - 3x \leq 4\).
3. Solve the left part of the compound inequality: \(-4 \leq 7 - 3x\). Subtract 7 from both sides to isolate the term with \(x\): \(-4 - 7 \leq -3x\), which simplifies to \(-11 \leq -3x\).
4. Divide both sides of the inequality \(-11 \leq -3x\) by \(-3\). Remember to reverse the inequality sign when dividing by a negative number: \(\frac{-11}{-3} \geq x\), which simplifies to \(\frac{11}{3} \geq x\) or \(x \leq \frac{11}{3}\).
5. Now solve the right part of the compound inequality: \(7 - 3x \leq 4\). Subtract 7 from both sides: \(-3x \leq 4 - 7\), which simplifies to \(-3x \leq -3\). Divide both sides by \(-3\), reversing the inequality sign: \(x \geq 1\). Combine this with the previous result to get the solution set: \(1 \leq x \leq \frac{11}{3}\).

Key Concepts

Absolute Value Inequalities

An absolute value inequality involves expressions within absolute value bars, representing distance from zero. To solve |A| ≤ B, where B ≥ 0, rewrite it as a compound inequality: -B ≤ A ≤ B. This approach helps isolate the variable and find the range of values satisfying the inequality.

Solving Linear Inequalities

Linear inequalities involve expressions with variables to the first power. Solving them requires isolating the variable by performing inverse operations, while remembering to reverse the inequality sign when multiplying or dividing by a negative number. The solution is often expressed as an interval or inequality.

Interval Notation

Interval notation is a concise way to represent sets of numbers between two endpoints. Square brackets [ ] indicate inclusion of endpoints, while parentheses ( ) indicate exclusion. It is commonly used to express solution sets of inequalities, showing all values that satisfy the condition.