Solve |(x+3)² - (x-3)²| = 5: Absolute Value Equations
Hey guys! Absolute value equations can seem tricky, but don't worry, we're going to break down how to solve one step-by-step. Specifically, we're tackling the equation |(x+3)² - (x-3)²| = 5. This guide will walk you through each stage, ensuring you understand not just the how, but also the why behind every move. So, grab a pen and paper, and let's get started!
Understanding Absolute Value
Before diving into the equation, let's quickly recap what absolute value means. The absolute value of a number is its distance from zero on the number line. This means that absolute value is always non-negative. For example, |3| = 3 and |-3| = 3. Understanding this fundamental concept is crucial for solving absolute value equations correctly.
When we encounter an equation like |something| = a, where a is a positive number, it implies that "something" can be either 'a' or '-a'. This is because both 'a' and '-a' are 'a' units away from zero. This principle forms the basis for solving our equation.
Absolute value ensures that regardless of whether the expression inside the absolute value bars evaluates to a positive or negative number, the result is always positive. This is why we need to consider both positive and negative scenarios when solving these types of equations. Consider the real-world implications; distance, for example, is always a positive quantity, regardless of direction. This concept mirrors the behavior of absolute value.
Therefore, when approaching an absolute value equation, always remember to split it into two separate equations, one where the expression inside the absolute value is equal to the positive value on the other side of the equation, and another where it is equal to the negative value. This approach ensures that you account for all possible solutions and accurately solve the equation. Keeping this in mind will prevent common errors and make the process much smoother. Let's move forward and apply this to our equation now!
Simplifying the Equation
Okay, let's simplify the expression inside the absolute value: (x+3)² - (x-3)². We'll start by expanding each squared term.
Remember the formula (a+b)² = a² + 2ab + b². Applying this, we get:
(x+3)² = x² + 6x + 9
Similarly, (a-b)² = a² - 2ab + b², so:
(x-3)² = x² - 6x + 9
Now, substitute these back into the original expression:
(x² + 6x + 9) - (x² - 6x + 9)
Distribute the negative sign:
x² + 6x + 9 - x² + 6x - 9
Notice that x² and -x² cancel out, and 9 and -9 also cancel out. This leaves us with:
6x + 6x = 12x
So, our original equation |(x+3)² - (x-3)²| = 5 simplifies to |12x| = 5. This is much easier to work with!
Solving for x
Now that we have simplified the equation to |12x| = 5, we can proceed to solve for x. Remember, the absolute value means that 12x could be either 5 or -5.
Case 1: 12x = 5
To solve for x, divide both sides by 12:
x = 5/12
Case 2: 12x = -5
Similarly, divide both sides by 12:
x = -5/12
Therefore, the solutions to the equation |(x+3)² - (x-3)²| = 5 are x = 5/12 and x = -5/12. These are the two values of x that satisfy the given equation. Always remember to check your answers by substituting them back into the original equation to ensure they are correct. In this case, substituting both values will confirm that they are indeed solutions.
Checking the Solutions
It's always a good idea to check our solutions to make sure they are correct. Let's plug x = 5/12 and x = -5/12 back into the original equation |(x+3)² - (x-3)²| = 5.
For x = 5/12:
|(5/12 + 3)² - (5/12 - 3)²| = |(5/12 + 36/12)² - (5/12 - 36/12)²| = |(41/12)² - (-31/12)²| = |(1681/144) - (961/144)| = |720/144| = |5| = 5
So, x = 5/12 is indeed a solution.
For x = -5/12:
|(-5/12 + 3)² - (-5/12 - 3)²| = |(-5/12 + 36/12)² - (-5/12 - 36/12)²| = |(31/12)² - (-41/12)²| = |(961/144) - (1681/144)| = |-720/144| = |-5| = 5
And x = -5/12 is also a solution. Great! Both solutions check out.
Conclusion
Alright, we've successfully solved the absolute value equation |(x+3)² - (x-3)²| = 5. We simplified the equation, solved for x, and checked our solutions. The solutions are x = 5/12 and x = -5/12. Remember the key steps:
- Simplify the expression inside the absolute value.
- Set up two equations: one where the expression equals the positive value and one where it equals the negative value.
- Solve each equation for x.
- Check your solutions by plugging them back into the original equation.
By following these steps, you can tackle similar absolute value equations with confidence. Keep practicing, and you'll become a pro in no time! Hope this helps, and happy solving! This structured approach ensures accuracy and avoids common mistakes in solving absolute value problems. Keep these principles in mind, and you'll find solving these equations much easier. Good luck, and keep practicing!