Solving Absolute Value Equations: A Step-by-Step Guide

Dec 4, 2025 by Alex Johnson 55 views

Have you ever encountered an equation with absolute values and felt a little lost? Don't worry, you're not alone! Absolute value equations can seem tricky at first, but with a clear understanding of the concepts and a step-by-step approach, you'll be solving them like a pro in no time. In this guide, we'll tackle the equation |(2a-5)/5| = 3 and break down the process into easy-to-follow steps. So, grab your pen and paper, and let's dive in!

Understanding Absolute Value

Before we jump into solving the equation, let's quickly review what absolute value means. The absolute value of a number is its distance from zero on the number line. It's always a non-negative value. Think of it like this: whether you're 5 steps to the left of zero (-5) or 5 steps to the right of zero (+5), your distance from zero is the same – 5. We represent absolute value using vertical bars, like this: |x|. So, |5| = 5 and |-5| = 5.

Why Absolute Value Matters in Equations

The key thing to remember about absolute value equations is that the expression inside the absolute value bars can be either positive or negative and still result in the same absolute value. For example, if |x| = 3, then x could be either 3 or -3, because both |3| and |-3| equal 3. This is crucial when solving absolute value equations because it means we often need to consider two separate cases.

Breaking Down the Equation |(2a-5)/5| = 3

Now, let's focus on our equation: |(2a-5)/5| = 3. This equation states that the absolute value of the expression (2a-5)/5 is equal to 3. Based on our understanding of absolute value, this means that the expression (2a-5)/5 can be either 3 or -3. This is where we split the problem into two separate cases:

Case 1: (2a-5)/5 = 3
Case 2: (2a-5)/5 = -3

By considering both positive and negative possibilities, we ensure that we find all possible solutions for 'a'. Failing to account for both cases is a common mistake when dealing with absolute value equations, so always remember to split the equation!

Solving Case 1: (2a-5)/5 = 3

Let's start with the first case: (2a-5)/5 = 3. Our goal here is to isolate 'a' on one side of the equation. To do this, we'll perform a series of algebraic operations, making sure to apply the same operation to both sides of the equation to maintain balance.

Multiply both sides by 5: This will eliminate the fraction. Multiplying both sides of the equation by 5, we get:
```
5 * [(2a-5)/5] = 5 * 3
```
This simplifies to:
```
2a - 5 = 15
```
Add 5 to both sides: This will isolate the term with 'a'. Adding 5 to both sides, we have:
```
2a - 5 + 5 = 15 + 5
```
Which simplifies to:
```
2a = 20
```
Divide both sides by 2: This will finally isolate 'a'. Dividing both sides by 2, we get:
```
2a / 2 = 20 / 2
```
Which gives us our first solution:
```
a = 10
```

So, one possible value for 'a' is 10. But remember, we have another case to consider!

Solving Case 2: (2a-5)/5 = -3

Now, let's tackle the second case: (2a-5)/5 = -3. We'll follow the same steps as before, but this time, we're dealing with a negative value on the right side of the equation.

Multiply both sides by 5: Just like in Case 1, we multiply both sides by 5 to eliminate the fraction:
```
5 * [(2a-5)/5] = 5 * (-3)
```
This simplifies to:
```
2a - 5 = -15
```
Add 5 to both sides: Adding 5 to both sides will isolate the term with 'a':
```
2a - 5 + 5 = -15 + 5
```
Which simplifies to:
```
2a = -10
```
Divide both sides by 2: Finally, we divide both sides by 2 to isolate 'a':
```
2a / 2 = -10 / 2
```
This gives us our second solution:
```
a = -5
```

So, the second possible value for 'a' is -5.

The Solutions: a = 10 and a = -5

We've successfully solved the absolute value equation |(2a-5)/5| = 3! By breaking it down into two cases and applying basic algebraic principles, we found two solutions:

a = 10
a = -5

These are the only two values of 'a' that will make the equation true. To be absolutely sure, you can always substitute these values back into the original equation and verify that they work.

Verifying the Solutions

It's always a good practice to check your solutions, especially with absolute value equations. Let's plug our values of 'a' back into the original equation |(2a-5)/5| = 3 to make sure they hold true.

Verifying a = 10

Substitute a = 10 into the equation:

|(2 * 10 - 5) / 5| = 3

Simplify the expression inside the absolute value bars:

|(20 - 5) / 5| = 3

|15 / 5| = 3

|3| = 3

3 = 3

This is true, so a = 10 is indeed a solution.

Verifying a = -5

Now, substitute a = -5 into the equation:

|(2 * (-5) - 5) / 5| = 3

Simplify the expression inside the absolute value bars:

|(-10 - 5) / 5| = 3

|-15 / 5| = 3

|-3| = 3

3 = 3

This is also true, confirming that a = -5 is a solution as well.

Key Takeaways for Solving Absolute Value Equations

Let's recap the key steps to solving absolute value equations like |(2a-5)/5| = 3:

Understand the Definition: Remember that absolute value represents distance from zero, so the expression inside the absolute value bars can be positive or negative.
Split into Two Cases: Create 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 one where it's equal to the negative value.
Solve Each Case: Use algebraic techniques to isolate the variable in each equation.
Verify Your Solutions: Plug your solutions back into the original equation to make sure they work.

By following these steps, you'll be well-equipped to tackle a wide range of absolute value equations.

Practice Makes Perfect

The best way to master solving absolute value equations is through practice. Try working through similar problems, and don't be afraid to make mistakes – they're a natural part of the learning process. The more you practice, the more confident you'll become in your ability to solve these types of equations.

Conclusion: You've Got This!

Solving absolute value equations might have seemed daunting at first, but hopefully, this step-by-step guide has demystified the process. Remember the core concepts, follow the steps, and you'll be solving equations like |(2a-5)/5| = 3 with ease. Keep practicing, and you'll be amazed at how quickly your skills develop. Now go forth and conquer those absolute values!

For further learning and practice, you might find resources on websites like Khan Academy's Algebra 1 section incredibly helpful.