Problem 71 - Algebra for College Students

Question

Solve each equation. \(|x-2|=|x+6|\)

Step-by-Step Solution

Answer

The solution is \(x = -2\).

Step 1: Understand Absolute Value Properties

The equation given is \(|x-2|=|x+6|\). Absolute value expressions \(|a|=|b|\) imply that either \(a = b\) or \(a = -b\). This means we have two separate cases to solve.

Step 2: Case 1 - Equal Values

For the first case, solve the equation \(x-2 = x+6\). By simplifying both sides, we subtract \(x\) from both sides to get \(-2 = 6\). This statement is false, meaning there is no solution in this case.

Step 3: Case 2 - Opposite Values

For the second case, solve \(x-2 = -(x+6)\). Distribute the negative sign to get \(x-2 = -x-6\). Combine like terms by adding \(x\) to both sides, yielding \(2x - 2 = -6\). Add 2 to both sides resulting in \(2x = -4\). Divide both sides by 2 to get \(x = -2\).

Step 4: Verify the Solution

Substitute \(x = -2\) back into the original equation to confirm: \(|-2-2| = |-2+6|\). This simplifies to \(|-4| = |4|\), which simplifies further to \(4 = 4\), confirming that the solution is correct.

Key Concepts

Solving Absolute Value Equations

The process of solving absolute value equations involves handling expressions that reflect the distance of a number from zero on the number line. When given an equation like \(|x-2|=|x+6|\), we need to consider what absolute value signifies. An absolute value equation \

• portrays the absolute values on each side, implying the possibility of the original expressions being equal or opposite in value.
• This results in two scenarios we must explore to find all possible solutions.

To solve such equations:
- Break down using the rule that \(|a| = |b|\) means either \(a = b\) or \(a = -b\).
- Create two separate equations from this logic and solve each equation individually.
In this exercise, we see that exploring both "equal" and "opposite" cases is key to finding correct and complete answers. This approach ensures that all potential solutions are uncovered.

Step-by-Step Solution Process

A systematic approach is crucial for solving equations effectively, particularly when dealing with absolute values. Each step helps to clarify the problem and validate the solution:

• First Case (Equal Values): Here, solve \(x-2 = x+6\).
  • Simplify both sides by eliminating the variable from one side and simplifying the integers, leading to a false statement.
  • This indicates that there is no solution for this scenario.
• Second Case (Opposite Values): Next, solve for \(x-2 = -(x+6)\).
  • Distribute the negative on the right side, followed by combining terms for simplification.
  • Through straightforward algebraic techniques, find \(x = -2\).
• Verification: Always substitute the potential solution back into the original equation to ensure correctness.
  • In this case, replacing \(x\) with \(-2\) yields a true statement, confirming the solution is accurate.

This methodical check confirms the thoroughness and validity of your solution.

Understanding Algebraic Concepts

When solving absolute value equations, understanding algebraic principles is vital. The concept of absolute value represents a foundational element in algebra. Here are a few key concepts involved:

• Absolute Value: Represents the distance of a number from zero, always yielding a non-negative result.
• Linear Equations: Such equations, even involving absolute values, require standard techniques like combining like terms and isolating variables.
• Equivalence in Equations: Recognizing that two expressions can yield equivalent absolute values helps frame the entire problem.

Understanding and applying these basics make handling seemingly complex equations straightforward. Mastery of these algebraic strategies enhances problem-solving skills and aids in tackling a wide range of mathematical challenges.