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Ch. 1 - Equations and Inequalities
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 1 - Equations and InequalitiesProblem 37
Chapter 2, Problem 37

Solve each inequality. Give the solution set in interval notation.
∣5−3x∣≤7| 5 - 3x | ≤ 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 \(B \geq 0\).
Apply this property to the inequality: write it as a compound inequality: \(-7 \leq 5 - 3x \leq 7\).
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 \(-12 \leq -3x\).
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 \(x \leq 4\).
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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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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.
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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.
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