Solving For V In 15 = |v| - 19 A Step By Step Guide

Hey guys! Let's dive into a fun math problem today. We're going to break down how to solve for v in the equation 15 = |v| - 19. This involves understanding absolute values and a bit of algebraic manipulation. Don't worry, we'll take it step by step, making it super clear and easy to follow. So, grab your thinking caps, and let's get started!

Understanding the Absolute Value

Before we jump into solving for v, let's quickly recap what absolute value means. The absolute value of a number is its distance from zero on the number line. Distance is always non-negative, which means the absolute value of a number is always positive or zero. We denote the absolute value using vertical bars: |x|. For instance, |3| = 3 because 3 is three units away from zero, and |-3| = 3 because -3 is also three units away from zero. The key takeaway here is that the absolute value strips away the sign of the number, leaving us with its magnitude.

When we encounter an equation involving an absolute value, we need to consider two possibilities: the expression inside the absolute value bars could be positive or negative. This is because both a positive number and its negative counterpart have the same absolute value. For example, if |v| = 5, then v could be either 5 or -5. Understanding this dual possibility is crucial for correctly solving equations with absolute values. Now that we've refreshed our understanding of absolute values, we're well-equipped to tackle the equation 15 = |v| - 19. The next step is to isolate the absolute value term, which will help us reveal the possible values of v. So, let's move on to the next section where we'll start manipulating the equation to get |v| by itself.

Isolating the Absolute Value

Alright, let's get to the nitty-gritty of solving our equation: 15 = |v| - 19. The first step in solving for v is to isolate the absolute value term, which in this case is |v|. To do this, we need to get |v| by itself on one side of the equation. We can achieve this by performing the same operation on both sides of the equation, maintaining the balance. Remember, whatever we do to one side, we must do to the other!

In our equation, 15 = |v| - 19, we notice that 19 is being subtracted from |v|. To undo this subtraction, we need to add 19 to both sides of the equation. This is a fundamental algebraic principle: we use inverse operations to isolate variables or terms. Adding 19 to both sides gives us:

15 + 19 = |v| - 19 + 19

Simplifying both sides, we get:

34 = |v|

Now, we have successfully isolated the absolute value term! This equation tells us that the absolute value of v is equal to 34. This is a significant step because it sets us up to find the possible values of v. Remember, the absolute value of a number is its distance from zero, so there are two numbers that are 34 units away from zero: 34 and -34. This is where we need to consider both possibilities to find all solutions for v. In the next section, we'll explore these two possibilities and determine the values of v that satisfy the original equation.

Determining Possible Values for v

Now that we've isolated the absolute value and have the equation |v| = 34, it's time to figure out what values of v actually make this true. Remember, the absolute value of a number is its distance from zero. So, we're looking for all numbers that are 34 units away from zero on the number line. There are two such numbers: 34 and -34.

Let's break it down. If v is 34, then |v| = |34| = 34, which satisfies our equation. On the other hand, if v is -34, then |v| = |-34| = 34, which also satisfies our equation. This is the crucial point about absolute value equations: there are often two solutions because both the positive and negative versions of a number have the same absolute value.

To be absolutely sure, let's plug these values back into our original equation, 15 = |v| - 19, to verify that they work. First, let's try v = 34:

15 = |34| - 19 15 = 34 - 19 15 = 15

This is true, so v = 34 is indeed a solution. Now, let's try v = -34:

15 = |-34| - 19 15 = 34 - 19 15 = 15

This is also true, confirming that v = -34 is also a solution. Therefore, we have found two solutions for v: 34 and -34. These are the only values that satisfy the equation 15 = |v| - 19. In the next section, we'll summarize our findings and present the solutions in the requested format.

Summarizing the Solutions

Okay, guys, we've done it! We've successfully solved for v in the equation 15 = |v| - 19. Let's recap the steps we took to arrive at our solution. First, we isolated the absolute value term by adding 19 to both sides of the equation, giving us |v| = 34. Then, we recognized that the absolute value equation has two possible solutions because both a number and its negative have the same absolute value. We considered both cases: v = 34 and v = -34. We verified that both of these values satisfy the original equation.

Therefore, the solutions for v are 34 and -34. When asked to provide multiple solutions separated by commas, we write our answer as:

v = 34, -34

This is the complete solution set for the equation 15 = |v| - 19. We've not only found the solutions but also understood the underlying principles of absolute value and algebraic manipulation that allowed us to solve the problem. Hopefully, this step-by-step guide has made the process clear and understandable. Remember, the key to solving equations with absolute values is to isolate the absolute value term and then consider both the positive and negative possibilities for the expression inside the absolute value bars. With practice, you'll become a pro at solving these types of equations!

Final Answer

In conclusion, the solutions for v in the equation 15 = |v| - 19 are:

v = 34, -34

We've walked through the entire process, from understanding absolute values to isolating the variable and verifying our solutions. Keep practicing, and you'll master these types of problems in no time! Remember, math can be fun when you break it down step by step. Keep up the great work, everyone!