Ch. 1 - Equations and Inequalities

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 1 - Equations and Inequalities

Problem 13

Chapter 2, Problem 13

Solve each equation. | (x - 4)/ 2| = 5

Verified step by step guidance

Recognize that the equation involves an absolute value: \(| \frac{x - 4}{2} | = 5\). Recall that \(|A| = B\) means \(A = B\) or \(A = -B\).

Set up two separate equations based on the definition of absolute value: \(\frac{x - 4}{2} = 5\) and \(\frac{x - 4}{2} = -5\).

Solve the first equation \(\frac{x - 4}{2} = 5\) by multiplying both sides by 2 to eliminate the denominator: \(x - 4 = 10\).

Add 4 to both sides of the first equation to isolate \(x\): \(x = 10 + 4\).

Solve the second equation \(\frac{x - 4}{2} = -5\) similarly by multiplying both sides by 2: \(x - 4 = -10\), then add 4 to both sides to find \(x = -10 + 4\).

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Key Concepts

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

Absolute Value Definition

The absolute value of a number represents its distance from zero on the number line, always as a non-negative value. For an expression |A| = B, where B ≥ 0, the solutions are A = B or A = -B. Understanding this is crucial for solving equations involving absolute values.

Solving Linear Equations

After removing the absolute value by considering both positive and negative cases, solving the resulting linear equations involves isolating the variable using inverse operations like addition, subtraction, multiplication, or division. This step finds the exact values of the variable.

Checking for Extraneous Solutions

When solving absolute value equations, it's important to verify each solution by substituting back into the original equation. This ensures no extraneous solutions, which can arise from the process of squaring or splitting the equation, are included in the final answer.

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