Ch.2 Solving Absolute Value Equations | Finding Center And Radius

Hey guys! Let's dive into the fascinating world of absolute value equations and how to express them in the form $|x - b| = c$. This form is super useful because it tells us a lot about the equation's solutions. Specifically, we're going to focus on finding the 'center' (b) and the 'radius' (c) of the solution set. We'll tackle scenarios with two solutions and one solution, making sure we understand the underlying concepts. So, grab your thinking caps, and let's get started!

Understanding Absolute Value Equations

Before we jump into specific problems, let's make sure we're all on the same page about what absolute value equations actually mean. The absolute value of a number is its distance from zero. Think of it as the number's magnitude, regardless of its sign. So, $|5| = 5$ and $|-5| = 5$. This concept is crucial for understanding how these equations work and finding their solutions.

An absolute value equation in the form $|x - b| = c$ represents all the values of x that are a distance c away from the point b on the number line. The value b is often referred to as the "center" because it's the midpoint of the solutions, and c is the "radius," representing the distance from the center to each solution. Understanding this geometric interpretation makes solving these equations much more intuitive. When we have two solutions, they are equidistant from the center, creating a symmetrical spread. This symmetry is the key to finding the values of b and c.

Now, let's consider the different scenarios we might encounter. If c is positive, we'll generally have two solutions because there are two points (one on each side of b) that are a distance c away from b. If c is zero, we'll have exactly one solution, which is b itself. And if c is negative, we'll have no solutions, since absolute value cannot be negative. It's like saying, "Find a number whose distance from zero is -3," which is impossible! These nuances are critical to successfully manipulating and solving absolute value equations.

Case b: Two Solutions: $x = -2$, $x = -32$

Alright, let's get our hands dirty with our first problem. We're given two solutions, x = -2 and x = -32, and our mission is to express the absolute value equation in the form $|x - b| = c$. The key here is to find the center (b) and the radius (c) of these two solutions. Think of it like finding the middle ground and the distance from that middle ground to each solution. This approach makes the process quite straightforward and intuitive.

First, let's find the center (b). The center is simply the midpoint between the two solutions. We can find the midpoint by averaging the two solutions. It's like finding the average of two numbers, which gives you the number exactly in the middle. So, we add the two solutions together and divide by 2:

b = (-2 + (-32)) / 2 = -34 / 2 = -17

So, our center b is -17. Now that we've pinpointed the center, it's like we've found the heart of our equation. Everything else revolves around this central value.

Next, we need to find the radius (c). The radius is the distance from the center to either of the solutions. It doesn't matter which solution we choose because they are both equidistant from the center. So, let's pick x = -2. To find the distance, we subtract the center from the solution and take the absolute value. Remember, distance is always positive, and the absolute value ensures we get a positive value:

c = |-2 - (-17)| = |-2 + 17| = |15| = 15

Therefore, our radius c is 15. We now know the distance from our center to each solution, giving us a clear picture of the equation's spread.

Now that we have both b and c, we can plug them into our absolute value equation form, $|x - b| = c$. Substituting b = -17 and c = 15, we get:

|x - (-17)| = 15

Simplifying this, we have:

|x + 17| = 15

And there you have it! We've successfully expressed the absolute value equation with the given solutions in the required form. This equation tells us that the distance between x and -17 is 15. If you were to plot this on a number line, you'd see the solutions -2 and -32 sitting exactly 15 units away from -17.

To double-check our work, we can plug our original solutions back into the equation to make sure they satisfy it. For x = -2:

|-2 + 17| = |15| = 15

And for x = -32:

|-32 + 17| = |-15| = 15

Both solutions work perfectly! This verification step is super important to ensure we haven't made any calculation errors along the way. It's like the final seal of approval on our solution, giving us confidence in our answer.

Case c: One Solution: $x = 23$

Now, let's move on to the next scenario where we have only one solution: x = 23. This case might seem a bit simpler at first, but it highlights a very important concept in absolute value equations. When we have only one solution, it tells us something very specific about the value of c. Remember, the absolute value equation $|x - b| = c$ has a unique solution only when c is zero. This is because the only number whose absolute value is zero is zero itself.

So, in this case, we know immediately that c must be 0. This simplifies our problem significantly because we now only need to find the value of b. The single solution x = 23 directly gives us the center b. When there's only one solution, that solution is the center. There's no need to average two points or calculate a midpoint; the solution is the bullseye.

Therefore, b = 23. We've found our center simply by recognizing that the single solution must be the central point when the radius is zero. This direct relationship between a single solution and the center makes this case a breeze to solve.

Now, we can plug b = 23 and c = 0 into our absolute value equation form, $|x - b| = c$, to get:

|x - 23| = 0

This equation is saying that the distance between x and 23 is zero. The only number that satisfies this condition is 23 itself, which aligns perfectly with our given solution. It's a neat, tidy solution that directly reflects the nature of absolute value equations with a single solution.

Again, let's check our work by substituting x = 23 into the equation:

|23 - 23| = |0| = 0

The solution checks out, confirming that our equation is correct. This simple check provides reassurance that we've applied the correct logic and calculations. When dealing with mathematical problems, this validation step is invaluable.

Key Takeaways and General Strategy

Let's take a moment to recap what we've learned and solidify a general strategy for tackling these types of problems. We've seen how to express absolute value equations in the form $|x - b| = c$ when given the solution set. The crucial steps involve finding the center (b) and the radius (c), and these steps differ slightly depending on whether we have two solutions or one solution.

For two solutions:

1. Find the center (b): Average the two solutions. This is the midpoint between the two points on the number line.
2. Find the radius (c): Calculate the distance from the center to either solution by subtracting the center from the solution and taking the absolute value. Remember, distance is always positive.
3. Substitute b and c into the form |x - b| = c: This gives you the absolute value equation.
4. Verify your answer: Plug the original solutions back into the equation to ensure they satisfy it. This is a critical step to catch any potential errors.

For one solution:

1. Recognize that c = 0: When there's only one solution, the radius is always zero.
2. The solution is the center (b): The single solution directly gives you the value of b.
3. Substitute b and c into the form |x - b| = c: This provides the absolute value equation.
4. Check your answer: Substitute the solution back into the equation to confirm its validity.

By following these steps, you'll be well-equipped to tackle a wide range of absolute value equation problems. Remember, the key is to understand the underlying concepts of center and radius and how they relate to the solutions.

In summary, guys, we've explored how to convert solutions of absolute value equations into the standard form $|x - b| = c$. By finding the center and radius, we can easily represent the equation and verify our answers. Keep practicing, and you'll become absolute value equation pros in no time!