How To Solve Absolute Value

Dianna Russini

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Apr 05, 2025

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How To Solve Absolute Value Equations and Inequalities

Absolute value might seem daunting at first, but with a structured approach, solving absolute value equations and inequalities becomes straightforward. This guide breaks down the process, offering clear explanations and examples to boost your understanding.

Understanding Absolute Value

The absolute value of a number is its distance from zero on the number line. Therefore, it's always non-negative. The symbol for absolute value is | |. For example:

• |5| = 5
• |-5| = 5
• |0| = 0

Solving Absolute Value Equations

The key to solving absolute value equations lies in understanding that the expression inside the absolute value bars can be either positive or negative. This leads to two separate equations.

Step-by-step guide:

1. Isolate the absolute value: Get the absolute value expression by itself on one side of the equation.

2. Create two equations: Set up two equations: one where the expression inside the absolute value is equal to the other side of the equation, and another where the expression is equal to the negative of the other side.

3. Solve each equation: Solve each equation separately.

4. Check your solutions: Substitute your solutions back into the original equation to ensure they are valid. Sometimes, solutions might be extraneous (meaning they don't actually work in the original equation).

Example:

Solve |x - 3| = 5

1. Isolated: The absolute value is already isolated.

2. Two equations:

   • x - 3 = 5
   • x - 3 = -5

3. Solve:

   • x - 3 = 5 => x = 8
   • x - 3 = -5 => x = -2

4. Check:

   • |8 - 3| = |5| = 5 (Correct)
   • |-2 - 3| = |-5| = 5 (Correct)

Therefore, the solutions are x = 8 and x = -2.

Solving Absolute Value Inequalities

Solving absolute value inequalities is similar, but with a crucial difference: the resulting inequalities depend on whether the inequality is "less than" or "greater than."

Case 1: |expression| < a

This inequality means the expression is between -a and a.

Example: |x + 2| < 4

This translates to: -4 < x + 2 < 4

Solving this compound inequality gives: -6 < x < 2

Case 2: |expression| > a

This inequality means the expression is either less than -a or greater than a.

Example: |x - 1| > 3

This translates to: x - 1 > 3 OR x - 1 < -3

Solving these inequalities gives: x > 4 OR x < -2

Common Mistakes to Avoid

• Forgetting the negative case: Always consider both the positive and negative cases when solving absolute value equations.
• Incorrectly handling inequalities: Remember the different approaches for "<" and ">" inequalities.
• Not checking solutions: Always verify your solutions by substituting them back into the original equation or inequality.

By following these steps and practicing regularly, you can master solving absolute value equations and inequalities. Remember to break down the problem, think carefully about the possible scenarios, and always check your answers.