Solving Absolute Value Equations: 5|x-3| - 2 = 18
Hey guys! Today, we're going to dive into solving an absolute value equation. Absolute value equations might seem a little tricky at first, but once you understand the basic principles, they become quite straightforward. We'll break down the equation 5|x-3| - 2 = 18 step-by-step, so you can confidently tackle similar problems in the future. So, let's put on our math hats and get started!
Understanding Absolute Value
Before we jump 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. This means that the absolute value of a number is always non-negative. For example, |3| = 3 and |-3| = 3. The absolute value function essentially strips away the sign of the number, leaving you with its magnitude.
- Key Concept: The absolute value |x| represents the distance of x from 0. It's always non-negative.
When dealing with equations involving absolute values, we need to consider two possibilities: the expression inside the absolute value can be either positive or negative. This is because both a positive number and its negative counterpart have the same absolute value. For instance, if |x| = 5, then x could be either 5 or -5.
Why Two Cases?
Think about it this way: the equation |x| = a (where a is a positive number) is essentially asking, "What numbers are 'a' units away from zero?" There are always two such numbers: one positive and one negative. This fundamental concept is the key to solving absolute value equations correctly.
Visualizing Absolute Value
Imagine a number line. The absolute value of a number is simply its distance from zero. For example, both 4 and -4 are 4 units away from zero, so |4| = 4 and |-4| = 4. When we solve absolute value equations, we're essentially finding the numbers that satisfy a given distance condition.
Step-by-Step Solution for 5|x-3| - 2 = 18
Now, let’s get down to business and solve the equation 5|x-3| - 2 = 18. We'll take it one step at a time, so it's super clear.
Step 1: Isolate the Absolute Value Term
The first thing we need to do is isolate the absolute value term. This means getting the part with the absolute value, which is |x-3| in this case, all by itself on one side of the equation. To do this, we'll use basic algebraic manipulations. Remember, our goal is to undo any operations that are affecting the absolute value term.
We start with: 5|x-3| - 2 = 18
To isolate the absolute value, we first need to get rid of the -2. We do this by adding 2 to both sides of the equation:
5|x-3| - 2 + 2 = 18 + 2
This simplifies to:
5|x-3| = 20
Now, the absolute value term is almost isolated. We have a 5 multiplied by the absolute value. To undo this multiplication, we'll divide both sides of the equation by 5:
(5|x-3|) / 5 = 20 / 5
This gives us:
|x-3| = 4
Great! Now we have the absolute value term isolated. This sets us up perfectly for the next crucial step: considering both positive and negative cases.
Step 2: Consider Both Positive and Negative Cases
This is where the magic happens! Remember that the expression inside the absolute value bars, (x-3) in our case, can be either positive or negative while still resulting in the same absolute value. Since |x-3| = 4, this means that (x-3) could be either 4 or -4. This gives us two separate equations to solve:
- Case 1: x - 3 = 4
- Case 2: x - 3 = -4
It's super important to consider both cases because each one might lead to a different solution. Ignoring one case will result in an incomplete answer, and we definitely don't want that!
Step 3: Solve Each Case Separately
Now that we have our two cases, we'll solve each one individually. This involves simple algebraic steps to isolate x in each equation.
Case 1: x - 3 = 4
To solve for x, we need to undo the subtraction of 3. We do this by adding 3 to both sides of the equation:
x - 3 + 3 = 4 + 3
This simplifies to:
x = 7
So, our first solution is x = 7. Awesome!
Case 2: x - 3 = -4
Similarly, we solve for x by adding 3 to both sides of the equation:
x - 3 + 3 = -4 + 3
This simplifies to:
x = -1
Our second solution is x = -1. We're on a roll!
Step 4: Check Your Solutions
Okay, we've got two potential solutions: x = 7 and x = -1. But before we declare victory, it's essential to check if these solutions actually work in the original equation. This is a crucial step to ensure we haven't made any mistakes along the way. Plugging our solutions back into the original equation will help us confirm their validity. Remember, absolute value equations can sometimes produce extraneous solutions (solutions that don't actually satisfy the original equation), so checking is a must!
Checking x = 7
Let's substitute x = 7 into the original equation: 5|x-3| - 2 = 18
5|7-3| - 2 = 18
5|4| - 2 = 18
5 * 4 - 2 = 18
20 - 2 = 18
18 = 18
Yep, x = 7 checks out! The equation holds true when we substitute x = 7. This gives us confidence that it's a valid solution.
Checking x = -1
Now, let's substitute x = -1 into the original equation:
5|-1-3| - 2 = 18
5|-4| - 2 = 18
5 * 4 - 2 = 18
20 - 2 = 18
18 = 18
Fantastic! x = -1 also checks out. The equation holds true when we substitute x = -1. This confirms that both of our solutions are valid.
Final Answer
After going through all the steps and verifying our solutions, we can confidently state the final answer. The solutions to the equation 5|x-3| - 2 = 18 are:
- x = 7
- x = -1
So, there you have it! We've successfully solved the absolute value equation. Remember, the key is to isolate the absolute value term, consider both positive and negative cases, solve each case separately, and always check your solutions. Keep practicing, and you'll become a pro at solving these types of equations!
Practice Problems
To really solidify your understanding, here are a few practice problems for you to try:
- |2x + 1| = 5
- 3|x - 4| + 2 = 11
- |5 - x| = 8
Work through these problems using the steps we discussed. Don't be afraid to make mistakes – that's how we learn! The more you practice, the more comfortable you'll become with absolute value equations.
Conclusion
Alright, guys, that wraps up our deep dive into solving the absolute value equation 5|x-3| - 2 = 18. We've covered the fundamental concept of absolute value, the importance of considering both positive and negative cases, and the step-by-step process for solving these types of equations. Remember, math is like a muscle – the more you exercise it, the stronger it gets. So, keep practicing, keep asking questions, and most importantly, have fun with it!
Solving absolute value equations might seem daunting at first, but by breaking them down into manageable steps, they become much less intimidating. Always remember to isolate the absolute value, consider both cases, solve each one individually, and double-check your answers. With these tools in your math toolkit, you'll be well-equipped to tackle any absolute value equation that comes your way. Keep up the great work, and I'll catch you in the next math adventure!