Solving |3x-9|+4 = 5x+9: Find Solution & Extraneous!

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Alright, guys, let's dive into solving this absolute value equation! We've got $|3x - 9| + 4 = 5x + 9$, and our mission is to find both the actual solution and that sneaky extraneous solution. Buckle up; it's gonna be a fun ride!

Understanding Absolute Value Equations

Before we even start crunching numbers, let's quickly recap what absolute value equations are all about. The absolute value of a number is its distance from zero. So, $|x|$ is $x$ if $x$ is positive or zero, and it's $-x$ if $x$ is negative. This means when we have an equation like $|3x - 9| + 4 = 5x + 9$, we need to consider two possible scenarios:

1. The expression inside the absolute value is positive or zero: In this case, $|3x - 9| = 3x - 9$.
2. The expression inside the absolute value is negative: In this case, $|3x - 9| = -(3x - 9)$.

Understanding these two scenarios is crucial because they will lead us to two different equations, and one of them might just give us that extraneous solution we're hunting for. Remember, an extraneous solution is a value that we get when solving the equation, but it doesn't actually satisfy the original equation. It's like a false positive – looks good on paper, but doesn't work in reality.

Step-by-Step Solution

Let's break down the solution step-by-step to make sure we don't miss anything. First, we need to isolate the absolute value term. So, we'll subtract 4 from both sides of the equation:

$|3x - 9| = 5x + 9 - 4$

$|3x - 9| = 5x + 5$

Now, we'll consider our two scenarios.

Scenario 1: The expression inside the absolute value is positive or zero

In this case, $3x - 9 ">= 0$, so $|3x - 9| = 3x - 9$. Our equation becomes:

$3x - 9 = 5x + 5$

Let's solve for $x$:

Subtract $3x$ from both sides:

$-9 = 2x + 5$

Subtract 5 from both sides:

$-14 = 2x$

Divide by 2:

$x = -7$

So, we have a potential solution: $x = -7$. But hold on! We need to check if this solution is valid by plugging it back into the original equation and also checking if it satisfies the condition $3x - 9 >= 0$.

Checking the condition:

$3(-7) - 9 >= 0$

$-21 - 9 >= 0$

$-30 >= 0$

This is false. So, $x = -7$ does not satisfy the condition $3x - 9 >= 0$, meaning it might be our extraneous solution.

Checking the original equation:

$|3(-7) - 9| + 4 = 5(-7) + 9$

$|-21 - 9| + 4 = -35 + 9$

$|-30| + 4 = -26$

$30 + 4 = -26$

$34 = -26$

This is definitely false. So, $x = -7$ is indeed an extraneous solution.

Scenario 2: The expression inside the absolute value is negative

In this case, $3x - 9 < 0$, so $|3x - 9| = -(3x - 9)$. Our equation becomes:

$-(3x - 9) = 5x + 5$

$-3x + 9 = 5x + 5$

Let's solve for $x$:

Add $3x$ to both sides:

$9 = 8x + 5$

Subtract 5 from both sides:

$4 = 8x$

Divide by 8:

$x = ".5"$

So, we have another potential solution: $x = ".5"$. Again, we need to check if this solution is valid by plugging it back into the original equation and also checking if it satisfies the condition $3x - 9 < 0$.

Checking the condition:

$3(".5") - 9 < 0$

$1.5 - 9 < 0$

$-7.5 < 0$

This is true. So, $x = ".5"$ satisfies the condition $3x - 9 < 0$.

Checking the original equation:

$|3(".5") - 9| + 4 = 5(".5") + 9$

$|1.5 - 9| + 4 = 2.5 + 9$

$|-7.5| + 4 = 11.5$

$7.5 + 4 = 11.5$

$11.5 = 11.5$

This is true. So, $x = ".5"$ is a valid solution.

Conclusion

After all that work, we've found that the equation $|3x - 9| + 4 = 5x + 9$ has one solution and one extraneous solution:

• Solution: $x = ".5"$
• Extraneous Solution: $x = -7$

Key Takeaways

• Always remember to check your solutions in the original equation, especially when dealing with absolute values or radicals.
• Extraneous solutions can arise when we perform operations that aren't reversible, like squaring both sides of an equation or, in this case, dealing with the two possible cases of an absolute value.
• Don't be afraid to break down the problem into smaller, manageable steps. It makes the whole process less daunting.

So there you have it! We successfully navigated the absolute value equation, identified the real solution, and unmasked the extraneous one. High five!

Why is it Important to Find Extraneous Solutions?

Now, you might be wondering, why all the fuss about extraneous solutions? Why bother checking if a solution is "real" or not? Well, the answer is quite simple: accuracy and reliability. In mathematics, and especially in real-world applications of math, getting the correct answer is paramount. Extraneous solutions, if not identified, can lead to incorrect conclusions, flawed designs, or even catastrophic failures.

Imagine, for instance, you're designing a bridge and using an equation to calculate the necessary strength of certain components. If you include an extraneous solution in your calculations, you might underestimate the required strength, leading to a bridge that collapses under its own weight. Okay, that's a dramatic example, but it illustrates the point.

In more abstract mathematical contexts, extraneous solutions can invalidate proofs or lead to incorrect generalizations. Therefore, being diligent about checking solutions and identifying extraneous ones is a fundamental aspect of mathematical rigor.

Common Mistakes to Avoid

When solving absolute value equations (or any equation, really), there are some common pitfalls that students often fall into. Here are a few to watch out for:

1. Forgetting to consider both cases: This is the biggest one. As we saw, absolute value equations require us to consider both the positive and negative cases of the expression inside the absolute value. If you only consider one case, you're likely to miss a solution (or incorrectly identify an extraneous one).
2. Not isolating the absolute value: Before splitting into cases, make sure the absolute value term is isolated on one side of the equation. Otherwise, you'll be working with incorrect expressions.
3. Incorrectly applying the negative sign: When dealing with the negative case, make sure you distribute the negative sign correctly to all terms inside the parentheses.
4. Skipping the check: We can't stress this enough: always check your solutions in the original equation! This is the only way to catch extraneous solutions.
5. Making algebraic errors: Simple mistakes in algebra can throw off your entire solution. Double-check your work at each step to minimize the risk of errors.

By being aware of these common mistakes, you can significantly improve your accuracy and confidence when solving absolute value equations.

Real-World Applications of Absolute Value Equations

While absolute value equations might seem like an abstract mathematical concept, they actually have a surprising number of real-world applications. Here are just a few examples:

1. Engineering: As mentioned earlier, engineering often involves calculations where precision is critical. Absolute value equations can be used to model tolerances, which define the acceptable range of variation in measurements or parameters. For example, a mechanical engineer might use an absolute value equation to specify the allowable deviation in the diameter of a manufactured part.
2. Physics: Absolute value is used extensively in physics to represent magnitudes (e.g., speed, distance) without regard to direction. For instance, the speed of an object is the absolute value of its velocity. Absolute value equations can also appear in various physics problems involving distance, time, and velocity.
3. Computer Science: In computer science, absolute value is used in various algorithms, such as those related to image processing, signal processing, and machine learning. For example, the absolute difference between two pixel values can be used to detect edges in an image.
4. Economics: Absolute value can be used to model deviations from a target value. For example, an economist might use an absolute value equation to analyze the fluctuations in a country's GDP around its projected growth rate.
5. Everyday Life: Even in everyday life, we encounter situations where absolute value is implicitly used. For example, when estimating the time it takes to travel somewhere, we're often concerned with the absolute difference between our estimated arrival time and our desired arrival time.

These are just a few examples, but they illustrate that absolute value equations are not just theoretical constructs. They are powerful tools that can be used to model and solve real-world problems in various fields.

By mastering the techniques for solving absolute value equations and understanding their applications, you'll be well-equipped to tackle a wide range of mathematical and real-world challenges. Keep practicing, and don't be afraid to ask questions! You got this!