Solving Absolute Value Equations A Step-by-Step Guide

$$

Hey everyone! Let's break down this absolute value equation step by step and find the solution together. Absolute value equations might seem tricky at first, but with a clear approach, they become much easier to handle. We're given the equation $2|4-5x|-8=-2$ and the possible solutions are:

A. $x=\frac{7}{5}$ B. No solution C. $x=-\frac{1}{5}, \frac{7}{5}$ D. $x=\frac{1}{5}, \frac{7}{5}$

Our goal is to determine which of these options is correct. Let’s dive in!

Understanding Absolute Value

Before we jump into solving, let's quickly recap what absolute value means. The absolute value of a number is its distance from zero on the number line. It's always non-negative. For example, $|3| = 3$ and $|-3| = 3$. This means that when we have an absolute value equation like $|x| = a$, we need to consider two cases: $x = a$ and $x = -a$. This is because both $a$ and $-a$ are the same distance from zero.

Step-by-Step Solution

Okay, let's solve the equation $2|4-5x|-8=-2$.

1. Isolate the Absolute Value Term

First things first, we need to isolate the absolute value term. This means getting the $|4-5x|$ part by itself on one side of the equation. To do this, we'll add 8 to both sides:

$2|4-5x|-8+8=-2+8$

This simplifies to:

$2|4-5x|=6$

Next, we divide both sides by 2:

$\frac{2|4-5x|}{2}=\frac{6}{2}$

Which gives us:

$|4-5x|=3$

Now we have the absolute value term isolated, which is crucial for the next step.

2. Set Up Two Equations

Remember, the absolute value means we have two possibilities to consider. The expression inside the absolute value can be either 3 or -3. So, we set up two separate equations:

• Case 1: $4-5x = 3$
• Case 2: $4-5x = -3$

3. Solve Case 1: $4-5x = 3$

Let's tackle the first case. We need to solve for $x$. First, subtract 4 from both sides:

$4-5x-4=3-4$

This simplifies to:

$-5x=-1$

Now, divide both sides by -5:

$\frac{-5x}{-5}=\frac{-1}{-5}$

So, we get:

$x=\frac{1}{5}$

4. Solve Case 2: $4-5x = -3$

Moving on to the second case, we have $4-5x = -3$. Again, subtract 4 from both sides:

$4-5x-4=-3-4$

This simplifies to:

$-5x=-7$

Now, divide both sides by -5:

$\frac{-5x}{-5}=\frac{-7}{-5}$

Which gives us:

$x=\frac{7}{5}$

5. Check the Solutions

It's always a good idea to check our solutions in the original equation to make sure they're valid. This is especially important with absolute value equations because sometimes we can get extraneous solutions (solutions that don't actually work).

Checking $x=\frac{1}{5}$

Plug $x=\frac{1}{5}$ into the original equation $2|4-5x|-8=-2$:

$2|4-5(\frac{1}{5})|-8=-2$

$2|4-1|-8=-2$

$2|3|-8=-2$

$2(3)-8=-2$

$6-8=-2$

$-2=-2$

This solution checks out!

Checking $x=\frac{7}{5}$

Now, let's check $x=\frac{7}{5}$:

$2|4-5(\frac{7}{5})|-8=-2$

$2|4-7|-8=-2$

$2|-3|-8=-2$

$2(3)-8=-2$

$6-8=-2$

$-2=-2$

This solution also checks out!

Final Answer

Alright, guys, we've solved the equation and checked our answers. We found two solutions: $x=\frac{1}{5}$ and $x=\frac{7}{5}$.

So, the correct answer is:

D. $x=\frac{1}{5}, \frac{7}{5}$

Great job working through this with me! Absolute value equations might seem tough, but breaking them down step by step makes them much more manageable. Remember to isolate the absolute value, set up two equations, solve each one, and always check your answers. Keep practicing, and you'll become a pro at these in no time!

Why This Approach Works

The key to solving absolute value equations is understanding the dual nature of absolute value. An absolute value expression $|A|$ can be equal to $A$ if $A$ is positive or zero, and it can be equal to $-A$ if $A$ is negative. This is why we split the original equation into two separate equations: one where the expression inside the absolute value is equal to the positive value on the other side, and one where it is equal to the negative value.

By isolating the absolute value term first, we ensure that we're applying this principle correctly. Then, by solving each case separately, we account for all possible solutions. Checking the solutions at the end is crucial because sometimes the process of solving can introduce solutions that don't actually satisfy the original equation. These are called extraneous solutions, and we want to make sure we exclude them.

Tips for Solving Absolute Value Equations

To make sure you ace every absolute value equation, here are a few more tips to keep in mind:

1. Isolate First: Always isolate the absolute value term before doing anything else. This is the most critical step.
2. Set Up Two Cases: Remember to create two separate equations, one for the positive case and one for the negative case.
3. Solve Each Case: Solve each equation independently.
4. Check Your Solutions: Always plug your solutions back into the original equation to check for extraneous solutions.
5. Be Careful with Signs: Pay close attention to signs, especially when distributing negatives or dealing with negative numbers inside the absolute value.
6. Practice Makes Perfect: The more you practice, the more comfortable you'll become with these types of equations.

Common Mistakes to Avoid

Avoiding mistakes is as important as knowing the correct method. Here are some common pitfalls to watch out for:

• Forgetting to Isolate: This is the biggest mistake. If you don't isolate the absolute value term first, you're likely to get incorrect solutions.
• Ignoring the Negative Case: Remember that the expression inside the absolute value can be negative, so you need to consider both the positive and negative cases.
• Not Checking Solutions: Extraneous solutions can sneak in, so always check your answers.
• Distributing Incorrectly: Be careful when distributing a negative sign in the negative case. Make sure you distribute it to all terms.
• Combining Before Isolating: Don't try to combine terms outside the absolute value with terms inside the absolute value until you've isolated the absolute value term.

Conclusion

In conclusion, guys, solving absolute value equations involves a clear process: isolate, split into two cases, solve, and check. By following these steps and avoiding common mistakes, you can confidently tackle any absolute value equation that comes your way. Remember to practice regularly, and you'll find these equations become much less daunting. Keep up the great work, and you'll master this skill in no time! Happy solving!