Solving Absolute Value Equations: A Step-by-Step Guide
Hey guys! Let's dive into the world of absolute value equations. Absolute value equations might seem tricky at first, but don't worry, we'll break it down step by step. This guide will help you understand how to solve them with confidence. We'll take a look at an example equation, |n-4|/5 = 9, and figure out how to find all the possible solutions. So, let’s get started and make solving these equations a breeze!
Understanding Absolute Value
First off, let's quickly recap what absolute value actually means. The absolute value of a number is its distance from zero on the number line. It’s always non-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. This is because both 5 and -5 are five units away from zero. Knowing this is crucial for tackling absolute value equations because it means there are usually two possibilities to consider.
When we're solving equations involving absolute values, we're essentially trying to find the numbers that, when plugged into the expression inside the absolute value bars, give us a certain distance from zero. This is why we end up with two separate equations to solve. It might seem a bit confusing now, but as we work through our example, it will all click into place. Think of absolute value as a way of stripping away the sign (positive or negative) and just focusing on the magnitude. This concept is key to understanding why we split the equation into two cases, one where the expression inside the absolute value is positive, and another where it's negative. It’s like we're covering all the bases to make sure we don't miss any solutions. So, with that in mind, let’s move on to our example equation and see how this works in practice.
The Equation: |n-4|/5 = 9
Let's get into our specific equation: |n-4|/5 = 9. This equation asks us to find the values of 'n' that make the expression true. Remember, the absolute value part, |n-4|, means we need to consider both the positive and negative scenarios of the expression inside the absolute value. Our main goal here is to isolate the absolute value expression first. This means we need to get |n-4| by itself on one side of the equation before we can split it into two separate cases. To do this, we'll need to undo the division by 5. Think of it like peeling back the layers of an onion; we're working our way from the outside in to get to the heart of the equation, which is the absolute value expression itself.
Once we've isolated the absolute value, we can then tackle the two possibilities: one where the expression inside the absolute value bars is positive and one where it's negative. This is the core strategy for solving these types of equations, and it's what allows us to account for both possible distances from zero. By isolating the absolute value first, we set ourselves up for a clearer path to finding all the solutions. So, let’s jump into the next step, where we'll actually start manipulating the equation to get that absolute value term all by its lonesome.
Step 1: Isolate the Absolute Value
To isolate the absolute value in the equation |n-4|/5 = 9, we need to get rid of the division by 5. To do this, we can multiply both sides of the equation by 5. This is a fundamental algebraic principle: what you do to one side of the equation, you must do to the other to maintain the balance. Multiplying both sides by 5 gives us: (|n-4|/5) * 5 = 9 * 5. On the left side, the 5 in the numerator and the 5 in the denominator cancel each other out, leaving us with |n-4|. On the right side, 9 multiplied by 5 equals 45. So, our equation now simplifies to: |n-4| = 45.
Great! Now we have the absolute value expression all by itself on one side of the equation. This is a crucial step because it sets us up to deal with the two possible scenarios that absolute value presents. Remember, the absolute value of something can be either positive or negative, but its distance from zero is always positive. So, now that we've isolated the absolute value, we can move on to the next stage, which involves splitting this single equation into two separate equations. This is where we'll consider both the positive and negative possibilities of what's inside the absolute value bars. Get ready, because this is where the real solution-finding begins!
Step 2: Split into Two Equations
Now that we have |n-4| = 45, we need to consider both scenarios: when the expression inside the absolute value, (n-4), is equal to 45, and when it's equal to -45. Remember, the absolute value of both 45 and -45 is 45, so both these cases are valid. This is the heart of solving absolute value equations – recognizing that the expression inside the absolute value can be either positive or negative and still satisfy the equation.
So, we create two separate equations:
- n - 4 = 45
- n - 4 = -45
By splitting the original equation into these two, we’re essentially saying,