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

Nov 14, 2025 by Alex Johnson 55 views

Welcome! Let's dive into the world of absolute value equations. Today, we'll tackle the equation $|2x + 3| = 7$ . Our goal is to find all the values of x that make this equation true. Understanding absolute values is key, so let's break it down in a way that's easy to grasp. We will discover the two solutions to this equation by following a clear, step-by-step approach. This will involve understanding what the absolute value means and how it affects the equation. We'll explore two separate cases, solving for x in each one, and finally, we will verify our answers to ensure they satisfy the original equation. We aim to make sure that by the end of this guide, you will confidently solve similar absolute value equations. Are you ready to begin? Let's get started!

Understanding Absolute Value: The Foundation

First, let's make sure we're all on the same page regarding absolute value. The absolute value of a number is its distance from zero on the number line. This distance is always positive or zero, no matter whether the original number is positive or negative. For example, the absolute value of 5, written as $|5|$ , is 5. And the absolute value of -5, written as $|-5|$ , is also 5. Think of it like a measuring tape; you're only concerned with the length, not the direction. This fundamental concept is crucial for understanding how to solve absolute value equations. When we see an equation like $|2x + 3| = 7$ , we're saying that the expression inside the absolute value, which is $(2x + 3)$ , is 7 units away from zero on the number line. Because it's a distance, it could be either in the positive or negative direction. This gives us two possibilities to consider when solving the equation. Knowing this will help us in the next step to figure out the right solutions. Remember, the absolute value eliminates the negative sign, so both positive and negative numbers can result in the same absolute value.

The Two Cases

Because the expression inside the absolute value can be either positive or negative, we have to split the absolute value equation into two separate cases. This is because both a positive and a negative version of the expression inside the absolute value can result in an absolute value of 7. Consider the equation $|2x + 3| = 7$ . This single equation actually represents two possibilities:

The expression inside the absolute value is positive: $2x + 3 = 7$
The expression inside the absolute value is negative: $2x + 3 = -7$

Each of these cases will lead to a different equation that we can solve to find the possible values of x. Solving these two equations will give us the two solutions to the original absolute value equation. This is the crucial step in solving any absolute value equation, because it breaks the equation into manageable parts. Now let's proceed to solve these equations step by step to find our answers. Remember that any absolute value equation will typically split into two separate equations.

Solving the First Case: Positive Scenario

Let's start with the first case, where the expression inside the absolute value is positive. This means we are solving the equation $2x + 3 = 7$ . To solve for x, we need to isolate it. Here's how:

Subtract 3 from both sides: This gives us $2x = 7 - 3$ , which simplifies to $2x = 4$ .
Divide both sides by 2: This isolates x, giving us $x = 4 / 2$ , which simplifies to $x = 2$ .

So, for the first case, the solution is $x = 2$ . This value of x makes the expression inside the absolute value equal to 7, satisfying the original equation. Let's keep this value in mind, since it's going to be one of our two solutions. We will verify this solution later to make sure we're on the right track. Remember, when you subtract or add from one side of an equation, you must do it on the other side as well to keep the equation balanced.

Verifying the First Solution

It's always a good idea to check your solution by substituting it back into the original equation. So, let's plug $x = 2$ back into the original equation $|2x + 3| = 7$ :

$|2(2) + 3| = 7$
$|4 + 3| = 7$
$|7| = 7$

Since $|7|$ is indeed equal to 7, our solution $x = 2$ is correct. It satisfies the original equation. This process is called verification, and it ensures that the solutions we found truly work. You should always take this step, because it can prevent simple calculation errors from ruining your results. If the result doesn't equal 7, you know you need to go back and check your work. Good job on finding one of our two solutions. We are ready for the next step. Let's move on to the next case to discover our second solution.

Solving the Second Case: Negative Scenario

Now, let's move on to the second case, where the expression inside the absolute value is negative. We're solving the equation $2x + 3 = -7$ . Here’s how we'll solve for x:

Subtract 3 from both sides: This gives us $2x = -7 - 3$ , which simplifies to $2x = -10$ .
Divide both sides by 2: This isolates x, giving us $x = -10 / 2$ , which simplifies to $x = -5$ .

Therefore, for the second case, the solution is $x = -5$ . This value of x makes the expression inside the absolute value equal to -7. Don't let the negative sign throw you off; the absolute value of -7 will result in 7, which fits the original equation. We'll proceed to verify this answer in the next section. Remember, each case gives us one potential solution, and it's essential to consider both the positive and negative scenarios when dealing with absolute values.

Verifying the Second Solution

Let's check our second solution by plugging $x = -5$ back into the original equation $|2x + 3| = 7$ :

$|2(-5) + 3| = 7$
$|-10 + 3| = 7$
$|-7| = 7$

Since the absolute value of -7 is 7, and 7 = 7, our solution $x = -5$ is correct. It also satisfies the original equation. Now, we have successfully found and verified both solutions to the absolute value equation. This demonstrates how to solve these types of equations completely. Remember, absolute value problems often have two possible answers, because of the concept of distance from zero.

Conclusion: Solutions to the Equation

So, to recap, the two solutions to the absolute value equation $|2x + 3| = 7$ are $x = 2$ and $x = -5$ . We arrived at these solutions by carefully considering both positive and negative scenarios for the expression inside the absolute value and then verifying each solution. We walked through each step meticulously. This method ensures we don't miss any possible solutions. Understanding absolute values and the two-case method is key to solving these types of equations. With practice, you will become comfortable and confident in solving them. Remember, always verify your answers to ensure they meet the original equation requirements. And now you know how to solve an absolute value equation. Congratulations on finishing this tutorial!

Key Takeaways:

Absolute Value: Understand that the absolute value of a number is its distance from zero, which is always positive or zero.
Two Cases: Always consider two cases when solving absolute value equations: one where the expression inside the absolute value is positive, and another where it's negative.
Solve and Verify: Solve for x in each case and then verify your solutions by substituting them back into the original equation.

By following these steps, you can confidently solve any absolute value equation. Keep practicing, and you'll master this concept in no time!

For further learning, you might find resources on solving different types of equations helpful. Here's a link to a comprehensive guide on equation solving:

Khan Academy - Solving Equations