Solving an absolute value equation involves a few clear steps. Let’s break them down to make the process easy to understand.
- Understand the Absolute Value Concept
The absolute value of a number is its distance from zero on the number line, regardless of direction. For example, $|3| = 3$ and $|-3| = 3$. This means that $|x| = a$ translates to $x = a$ or $x = -a$
- Set Up Two Separate Equations
When you have an absolute value equation like
$|x| = a$, you need to consider both the positive and negative scenarios. This means you will create two separate equations to solve:- $x = a$
- $x = -a$
Example
Consider the equation $|x – 5| = 3$
- $x – 5 = 3$
- $x – 5 = -3$
- Solve Each Equation Separately
Next, solve each equation as you would any linear equation.
For $x – 5 = 3$
Add 5 to both sides:
$x – 5 + 5 = 3 + 5$
$x = 8$
For $x – 5 = -3$
Add 5 to both sides:
$x – 5 + 5 = -3 + 5$
$x = 2$
So, the solutions are $x = 8$ and $x = 2$
- Check Your Solutions
It’s always good practice to check your solutions by plugging them back into the original equation to ensure they satisfy the equation.
For $x = 8$
$|8 – 5| = |3| = 3$ (Correct)
For $x = 2$
$|2 – 5| = |-3| = 3$ (Correct)
Both solutions are correct.
Special Cases
No Solution
Sometimes, an absolute value equation has no solution. For example, $|x| = -3$ has no solution because absolute values cannot be negative.
Extraneous Solutions
Occasionally, you might find extraneous solutions, especially in more complex equations. Always check your solutions to ensure they are valid.
Conclusion
Solving absolute value equations is straightforward once you understand the concept and follow these steps. First, break the absolute value equation into two separate linear equations. Then, solve each equation and check your solutions. By practicing these steps, you’ll become more confident in handling absolute value equations.