Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

1. Equations & Inequalities

Linear Inequalities

College Algebra

1. Equations & Inequalities

Linear Inequalities

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0:59 minutes

Problem 56

Textbook Question

Solve each equation or inequality. | 12- 9x | ≥ -12

Verified step by step guidance

Recognize that the inequality involves an absolute value: \(|12 - 9x| \geq -12\).

Understand that an absolute value expression is always non-negative, meaning \(|12 - 9x|\) is always greater than or equal to 0.

Since \(-12\) is less than 0, the inequality \(|12 - 9x| \geq -12\) is always true for any real number \(x\).

Conclude that the solution to the inequality is all real numbers, because the absolute value expression is always greater than or equal to any negative number.

Express the solution set as \(x \in \mathbb{R}\), indicating that any real number satisfies the inequality.

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Key Concepts

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

Absolute Value

Absolute value measures the distance of a number from zero on the number line, regardless of direction. It is denoted by vertical bars, such as |x|, and is always non-negative. For example, |3| = 3 and |-3| = 3. Understanding absolute value is crucial for solving equations and inequalities that involve it, as it can lead to two separate cases based on the definition.

Inequalities

Inequalities express a relationship between two expressions that are not necessarily equal, using symbols like ≥, ≤, >, or <. In this context, the inequality |12 - 9x| ≥ -12 indicates that the expression inside the absolute value is either greater than or equal to -12 or less than or equal to 12. Recognizing how to manipulate and solve inequalities is essential for finding valid solutions.

Case Analysis

When solving absolute value equations or inequalities, case analysis is a method used to break down the problem into simpler parts. For the inequality |12 - 9x| ≥ -12, we consider the two scenarios: when the expression inside the absolute value is non-negative and when it is negative. This approach allows us to solve for x in a structured manner, ensuring all possible solutions are considered.