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Chapter 2: Problem 454

Solve. $$ |6 x-5|=|2 x+3| $$

Short Answer

Expert verified
x = 2 and x = \frac{1}{4}

Step by step solution

01

Understand the Absolute Value Equation

Absolute value equations often have two cases because absolute value measures the distance from zero without regard to direction. For an equation |A| = |B|, this splits into two smaller equations: A = B and A = -B.
02

Set Up the Two Cases

Given the equation \(|6x - 5| = |2x + 3|\), we set up two cases. Case 1: \(6x - 5 = 2x + 3\) and Case 2: \(6x - 5 = -(2x + 3)\).
03

Solve Case 1

For Case 1: \(6x - 5 = 2x + 3\), subtract 2x from both sides to get \(4x - 5 = 3\). Then add 5 to both sides to find \(4x = 8\). Finally, divide both sides by 4 to get \(x = 2\).
04

Solve Case 2

For Case 2: \(6x - 5 = -(2x + 3)\). Simplify the right-hand side to get \(6x - 5 = -2x - 3\). Then, add 2x to both sides to get \(8x - 5 = -3\). Next, add 5 to both sides to find \(8x = 2\). Finally, divide both sides by 8 to get \(x = \frac{1}{4}\).
05

Write the Final Solution

Combine the solutions from both cases. The solutions to the equation \(|6x - 5| = |2x + 3|\) are \(x = 2\) and \(x = \frac{1}{4}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

solving equations
you need to understand the nature of absolute values. Absolute values measure the distance from zero. They are always non-negative. This unique property means that each absolute value equation may have multiple solutions.

By understanding the given equation and correctly breaking it down step-by-step, you can find all possible values for the variable that satisfy the equation.

absolute value
@we set up two separate cases to analyze.
algebraic manipulation
@Carefully applying these operations is essential for solving for the variable accurately.
case analysis
After completing case analysis, review your work to ensure all steps are correctly followed. This practice helps to avoid errors and confirm that all potential solutions are considered.

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