Solving |3 + 10x| = 23 A Step-by-Step Guide

by Sharif Sakr 44 views

Hey guys! Let's dive into solving an absolute value equation. Absolute value equations might seem a bit tricky at first, but once you understand the underlying principle, they become quite straightforward. Today, we're going to tackle the equation |3 + 10x| = 23. We'll break down the steps, explain the logic, and arrive at the correct solutions. So, buckle up and let's get started!

Understanding Absolute Value

Before we jump into the solution, let's quickly recap what absolute value means. The absolute value of a number is its distance from zero on the number line. This distance is always non-negative. For instance, the absolute value of 5 (written as |5|) is 5, and the absolute value of -5 (written as |-5|) is also 5. The absolute value essentially strips away the sign, leaving you with the magnitude of the number.

When we have an equation involving an absolute value, like our |3 + 10x| = 23, it means that the expression inside the absolute value bars (in this case, 3 + 10x) can be either 23 or -23 because both 23 and -23 are 23 units away from zero. This is the key concept to grasp when solving absolute value equations. It's like saying, "Hey, whatever is inside these absolute value bars, it could be this positive number, or it could be this negative number, and both would still make the equation true because of the absolute value's effect."

This leads us to the fundamental principle for solving absolute value equations: We need to consider two separate cases. First, we'll assume the expression inside the absolute value bars is equal to the positive value on the other side of the equation. Then, we'll assume the expression is equal to the negative value. By solving both cases, we capture all possible solutions for the equation. This approach ensures that we don't miss any potential answers that satisfy the original equation. Think of it as covering all our bases – we're making sure we account for both scenarios where the expression inside the absolute value could lead to the given result.

So, with this understanding of absolute value in mind, we can confidently approach the equation |3 + 10x| = 23 by splitting it into two distinct cases. This is where the real problem-solving begins, and it's where we'll see how the concept of absolute value translates into concrete algebraic steps.

Breaking Down the Equation |3 + 10x| = 23

Okay, let's get to the core of the problem. As we discussed, the equation |3 + 10x| = 23 tells us that the expression 3 + 10x is either 23 units away from zero in the positive direction or 23 units away from zero in the negative direction. This gives us two possibilities to explore, and we'll handle each one separately.

Case 1: 3 + 10x = 23

In this case, we assume that the expression inside the absolute value bars, 3 + 10x, is equal to the positive value, 23. This gives us a straightforward linear equation to solve. Our goal is to isolate the variable x, so we'll use standard algebraic techniques to achieve that.

The first step is to get rid of the constant term, 3, on the left side of the equation. We can do this by subtracting 3 from both sides. Remember, whatever we do to one side of an equation, we must do to the other to maintain the balance. So, subtracting 3 from both sides of 3 + 10x = 23 gives us:

3 + 10x - 3 = 23 - 3

Simplifying this, we get:

10x = 20

Now, we have 10 times x equals 20. To isolate x, we need to undo the multiplication by 10. We can do this by dividing both sides of the equation by 10:

10x / 10 = 20 / 10

This simplifies to:

x = 2

So, the first possible solution we've found is x = 2. This means that if we plug x = 2 back into the original equation, the expression inside the absolute value should indeed equal 23 (or -23, which the absolute value will then turn into 23). We'll verify this later to ensure our solution is correct.

Case 2: 3 + 10x = -23

Now, let's consider the second possibility: that the expression inside the absolute value bars, 3 + 10x, is equal to the negative value, -23. This is equally important because the absolute value of -23 is also 23, which satisfies our original equation. This case will likely give us a different solution for x, and that's exactly what we need to find all possible answers.

Just like in Case 1, we'll start by isolating the x term. We begin by subtracting 3 from both sides of the equation 3 + 10x = -23:

3 + 10x - 3 = -23 - 3

Simplifying this, we get:

10x = -26

Next, we need to isolate x by dividing both sides of the equation by 10:

10x / 10 = -26 / 10

This simplifies to:

x = -26/10

We can further simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

x = -13/5

So, our second possible solution is x = -13/5. This means that if we plug x = -13/5 back into the original equation, the expression inside the absolute value should equal -23 (which, when we take the absolute value, becomes 23).

Verifying the Solutions

Alright, we've found two potential solutions: x = 2 and x = -13/5. But before we declare victory, it's crucial to verify that these solutions actually work. This is a critical step in solving any equation, but especially absolute value equations, as it helps us catch any potential errors we might have made along the way.

Verifying x = 2

Let's start with x = 2. We'll substitute this value back into the original equation, |3 + 10x| = 23, and see if it holds true.

Replacing x with 2, we get:

|3 + 10(2)| = 23

Now, we simplify the expression inside the absolute value bars:

|3 + 20| = 23

|23| = 23

And indeed, the absolute value of 23 is 23. So, x = 2 is a valid solution.

Verifying x = -13/5

Now, let's verify the second solution, x = -13/5. Again, we'll substitute this value back into the original equation and check if it satisfies the equation.

Replacing x with -13/5, we get:

|3 + 10(-13/5)| = 23

Simplifying the expression inside the absolute value bars:

|3 - 26| = 23

|-23| = 23

And the absolute value of -23 is indeed 23. So, x = -13/5 is also a valid solution.

Final Solutions and Conclusion

Great! We've found two solutions, and we've verified that both of them satisfy the original equation. That means we're confident in our answer. The solutions to the equation |3 + 10x| = 23 are:

  • x = 2
  • x = -13/5

Therefore, the correct answer is B. x = 2 and x = -13/5

In conclusion, solving absolute value equations involves considering two separate cases: one where the expression inside the absolute value bars is equal to the positive value on the other side of the equation, and another where it's equal to the negative value. By solving both cases and verifying our solutions, we can confidently find all possible answers. Absolute value equations might seem intimidating at first, but with a systematic approach and a clear understanding of the definition of absolute value, you can conquer them with ease. Keep practicing, and you'll become an absolute value equation-solving pro in no time!