Solving Absolute Value Equations: |3x - 9| = 9

by Alex Johnson 47 views

Absolute value equations might seem tricky at first, but they're actually quite straightforward once you understand the basic principle. This article will walk you through the process of solving the equation |3x - 9| = 9, breaking down each step to ensure you grasp the concept thoroughly. Whether you're a student tackling algebra or just brushing up on your math skills, this guide will provide a clear and concise explanation.

Understanding Absolute Value

Before we dive into solving 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. Distance is always non-negative, so the absolute value of a number is always positive or zero. We denote the absolute value using vertical bars, like this: |x|. For example:

  • |5| = 5, because 5 is 5 units away from zero.
  • |-5| = 5, because -5 is also 5 units away from zero.
  • |0| = 0, because 0 is 0 units away from zero.

Key takeaway: Absolute value essentially strips away the negative sign, giving you the magnitude of the number.

The Implications for Equations

Now, consider an equation like |x| = 3. This equation is asking: "What numbers have a distance of 3 from zero?" There are two such numbers: 3 and -3. This is a fundamental concept when solving absolute value equations – you need to consider both the positive and negative possibilities.

When dealing with more complex expressions inside the absolute value bars, such as |3x - 9|, this principle still applies. The expression inside the bars, 3x - 9 in our case, can be equal to either the positive or the negative value on the other side of the equation.

Solving |3x - 9| = 9: A Step-by-Step Guide

Now that we've refreshed our understanding of absolute value, let's tackle the equation |3x - 9| = 9. Here's a step-by-step approach:

Step 1: Set up Two Equations

The absolute value equation |3x - 9| = 9 implies that the expression inside the absolute value bars, 3x - 9, can be equal to either 9 or -9. This is because both 9 and -9 have an absolute value of 9. Therefore, we need to create two separate equations:

  • Equation 1: 3x - 9 = 9
  • Equation 2: 3x - 9 = -9

This is the most crucial step in solving absolute value equations. By splitting the original equation into two, we account for both possible scenarios that satisfy the absolute value condition. If you only consider one case, you'll miss a solution.

Step 2: Solve Equation 1 (3x - 9 = 9)

Let's solve the first equation, 3x - 9 = 9. This is a simple linear equation, and we can solve it using basic algebraic manipulations.

  1. Add 9 to both sides: 3x - 9 + 9 = 9 + 9 3x = 18
  2. Divide both sides by 3: 3x / 3 = 18 / 3 x = 6

So, the first solution is x = 6. This means that when x is 6, the expression 3x - 9 equals 9, and the absolute value of 9 is indeed 9.

Step 3: Solve Equation 2 (3x - 9 = -9)

Now, let's solve the second equation, 3x - 9 = -9. Again, we'll use algebraic manipulations to isolate x.

  1. Add 9 to both sides: 3x - 9 + 9 = -9 + 9 3x = 0
  2. Divide both sides by 3: 3x / 3 = 0 / 3 x = 0

Therefore, the second solution is x = 0. This means that when x is 0, the expression 3x - 9 equals -9, and the absolute value of -9 is also 9.

Step 4: Check Your Solutions

It's always a good practice to check your solutions to make sure they are correct. This is especially important with absolute value equations, as extraneous solutions (solutions that don't actually satisfy the original equation) can sometimes arise.

  1. Check x = 6: |3(6) - 9| = |18 - 9| = |9| = 9. This solution is valid.
  2. Check x = 0: |3(0) - 9| = |0 - 9| = |-9| = 9. This solution is also valid.

Both solutions check out, so we're confident in our answer.

The Solution

The solutions to the equation |3x - 9| = 9 are x = 6 and x = 0. This corresponds to option (d) in the original problem.

Why Two Solutions?

Remember, the absolute value represents distance from zero. The equation |3x - 9| = 9 is asking for all values of x that make the expression 3x - 9 nine units away from zero. There are two such values: 9 and -9. This is why we get two solutions.

Common Mistakes to Avoid

Solving absolute value equations is fairly straightforward, but there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them.

Forgetting the Negative Case

The most common mistake is forgetting to consider the negative case. When you have an equation like |expression| = value, remember that the expression inside the absolute value can be equal to either the positive or the negative of the value. Always set up two equations to account for both possibilities.

Incorrectly Applying Operations Inside Absolute Value Bars

You cannot simply distribute or perform operations across the absolute value bars as you would with parentheses. The absolute value must be isolated first before you split the equation into two cases. For example, you can't change |3x - 9| to |3x| - |9|.

Not Checking for Extraneous Solutions

While not always necessary in simple absolute value equations like this one, it's a good habit to check your solutions, especially in more complex problems. Extraneous solutions can sometimes arise due to the nature of absolute value, so checking ensures your answers are valid.

Practice Problems

To solidify your understanding, try solving these practice problems:

  1. |2x + 1| = 5
  2. |4x - 8| = 0
  3. |x + 3| = 2

Work through these problems using the step-by-step method outlined above. Remember to set up two equations, solve each one, and check your solutions. The more you practice, the more comfortable you'll become with solving absolute value equations.

Real-World Applications of Absolute Value

Absolute value isn't just an abstract mathematical concept; it has real-world applications in various fields. Here are a few examples:

Error Measurement

In science and engineering, absolute value is often used to represent the magnitude of an error or deviation. For example, if you're measuring the length of an object and your measurement is off by 0.5 cm, the absolute value of the error is |0.5| = 0.5 cm, regardless of whether you over- or underestimated the length.

Distance Calculations

As we've discussed, absolute value represents distance from zero. This concept extends to calculating distances between any two points on a number line. The distance between two points a and b is given by |a - b|.

Computer Programming

In computer programming, absolute value functions are used in various algorithms, such as those involving distance calculations, error handling, and signal processing.

Finance

Absolute value can be used to represent the magnitude of a financial gain or loss, without regard to whether it's a profit or a loss. For example, a loss of $100 and a gain of $100 both have an absolute value of $100.

Conclusion

Solving absolute value equations involves understanding the concept of absolute value as distance from zero and applying a systematic approach. By splitting the equation into two cases (positive and negative), solving each equation separately, and checking your solutions, you can confidently tackle these types of problems. Remember to avoid common mistakes, such as forgetting the negative case or incorrectly applying operations inside the absolute value bars.

With practice, you'll master the art of solving absolute value equations and gain a valuable skill for your mathematical journey. Don't hesitate to revisit this guide and work through the examples and practice problems as needed. Keep practicing, and you'll be solving absolute value equations like a pro in no time!

For further learning on absolute values, check out this resource on Khan Academy.