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how to solve two absolute value equations

how to solve two absolute value equations

2 min read 05-02-2025
how to solve two absolute value equations

Solving equations with two absolute value expressions requires a methodical approach. Unlike single absolute value equations, these problems often lead to multiple solutions, demanding careful consideration of all possible scenarios. This guide will walk you through the process, explaining the logic and providing examples to solidify your understanding.

Understanding Absolute Value

Before diving into the solutions, let's refresh our understanding of absolute value. The absolute value of a number is its distance from zero on the number line. Therefore, it's always non-negative. For example:

  • |5| = 5
  • |-5| = 5

This means that if |x| = a, then x = a or x = -a. This principle forms the basis of solving absolute value equations.

Methods for Solving Two Absolute Value Equations

There are two primary methods for tackling equations with two absolute values:

Method 1: Case-by-Case Analysis

This method involves considering all possible combinations of positive and negative values within the absolute value expressions. Let's illustrate with an example:

Example: Solve |x - 2| = |2x + 1|

Step 1: Identify the Cases

We have two absolute value expressions, so there are four possible cases:

  • Case 1: (x - 2) ≥ 0 and (2x + 1) ≥ 0 => x - 2 = 2x + 1
  • Case 2: (x - 2) ≥ 0 and (2x + 1) < 0 => x - 2 = -(2x + 1)
  • Case 3: (x - 2) < 0 and (2x + 1) ≥ 0 => -(x - 2) = 2x + 1
  • Case 4: (x - 2) < 0 and (2x + 1) < 0 => -(x - 2) = -(2x + 1)

Step 2: Solve Each Case

Now we solve each equation individually:

  • Case 1: x - 2 = 2x + 1 => x = -3. However, this contradicts our initial assumption (x - 2 ≥ 0), so this solution is invalid.
  • Case 2: x - 2 = -2x - 1 => 3x = 1 => x = 1/3. This satisfies the conditions (x - 2 ≥ 0 is false, but 2x + 1 < 0 is true).
  • Case 3: -x + 2 = 2x + 1 => 3x = 1 => x = 1/3. This satisfies the conditions (x - 2 < 0 is false, but 2x + 1 > 0 is true).
  • Case 4: -x + 2 = -2x - 1 => x = -3. This satisfies the conditions (x - 2 < 0 and 2x + 1 < 0 are both true).

Step 3: Combine Solutions

We have two valid solutions: x = 1/3 and x = -3.

Method 2: Squaring Both Sides

This method involves squaring both sides of the equation to eliminate the absolute value symbols. However, this method requires careful attention to potential extraneous solutions (solutions that don't satisfy the original equation).

Example: Solve |x - 2| = |2x + 1| (same as above)

Step 1: Square Both Sides

(x - 2)² = (2x + 1)²

Step 2: Expand and Simplify

x² - 4x + 4 = 4x² + 4x + 1

3x² + 8x - 3 = 0

Step 3: Solve the Quadratic Equation

(3x - 1)(x + 3) = 0

x = 1/3 or x = -3

Step 4: Check for Extraneous Solutions

Substitute both solutions back into the original equation to verify they are valid. In this case, both solutions are valid.

Choosing the Best Method

Both methods are valid. The case-by-case analysis is often more straightforward for beginners, while squaring both sides can be faster for more complex equations, but requires careful checking for extraneous solutions.

Practice Problems

Try solving these equations using either method:

  1. |x + 1| = |2x - 3|
  2. |3x - 1| = |x + 5|
  3. |x - 4| = |2 - x|

Remember to always check your solutions by substituting them back into the original equation! Mastering absolute value equations takes practice. Work through several examples to build your confidence and skill.

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