Solve Absolute Value Equation

Algebra-2

1. Fundamental Concepts

Definition: An absolute value equation is an equation that contains an absolute value expression.
Absolute Value: The absolute value of a number is its distance from zero on the number line, always non-negative.
Solving Strategy: To solve an absolute value equation, isolate the absolute value expression and set up two separate equations: one with the positive value and one with the negative value.

2. Key Concepts

Basic Rule: $$|x| = a \implies x = a \text{ or } x = -a$$

Degree Preservation: The solution to an absolute value equation can have at most two solutions.

Application: Used in various real-world scenarios such as physics, engineering, and economics for modeling distances and tolerances.

3. Examples

Example 1 (Basic)

Problem: Solve $$|2x + 3| = 7$$

Step-by-Step Solution:

Set up two equations: $$2x + 3 = 7 \text{ and } 2x + 3 = -7$$
Solve each equation:
- For $$2x + 3 = 7$$: Subtract 3 from both sides to get $$2x = 4$$. Divide by 2 to get $$x = 2$$.
- For $$2x + 3 = -7$$: Subtract 3 from both sides to get $$2x = -10$$. Divide by 2 to get $$x = -5$$.

Validation: Substitute $$x = 2$$ and $$x = -5$$ into the original equation to verify.

Example 2 (Intermediate)

Problem: Solve $$|3x - 4| + 2 = 8$$

Step-by-Step Solution:

Isolate the absolute value expression: Subtract 2 from both sides to get $$|3x - 4| = 6$$.
Set up two equations: $$3x - 4 = 6 \text{ and } 3x - 4 = -6$$
Solve each equation:
- For $$3x - 4 = 6$$: Add 4 to both sides to get $$3x = 10$$. Divide by 3 to get $$x = \frac{10}{3}$$.
- For $$3x - 4 = -6$$: Add 4 to both sides to get $$3x = -2$$. Divide by 3 to get $$x = -\frac{2}{3}$$.

Validation: Substitute $$x = \frac{10}{3}$$ and $$x = -\frac{2}{3}$$ into the original equation to verify.

4. Problem-Solving Techniques

Isolation Technique: Always start by isolating the absolute value expression on one side of the equation.
Case Analysis: After isolation, consider both the positive and negative cases separately.
Verification Step: Always check your solutions by substituting them back into the original equation.