Solving Absolute Value Equations: 10 = |2x - 12|

Hey guys! Today, we're diving into the world of absolute value equations with a specific problem: 10 = |2x - 12|. Don't worry, it's not as intimidating as it looks. We'll break it down step-by-step, so you can confidently solve similar problems in the future. Understanding absolute value is key to acing these types of questions, so let's jump right in and get our hands dirty with some math!

Understanding Absolute Value

Before we tackle the equation, let's quickly recap what absolute value actually means. In simple terms, the absolute value of a number is its distance from zero on the number line. This distance is always non-negative. So, whether you're dealing with a positive number or a negative number, its absolute value will always be positive or zero.

Think of it like this: |5| = 5 because 5 is 5 units away from zero. Similarly, |-5| = 5 because -5 is also 5 units away from zero. The absolute value bars, those two vertical lines, are the key indicators that we're dealing with absolute value.

This concept is crucial when solving equations involving absolute values because it means there are usually two possible scenarios to consider. In our case, the expression inside the absolute value, (2x - 12), could equal either 10 or -10, since both of those numbers have an absolute value of 10. This is the core idea we'll use to solve the equation.

Setting Up the Two Equations

Now that we've got a solid grasp of absolute value, let's get back to our equation: 10 = |2x - 12|. Because of the absolute value, we need to consider two separate equations:

1. The first possibility is that the expression inside the absolute value, (2x - 12), is equal to 10. So, our first equation is: 2x - 12 = 10.
2. The second possibility is that the expression (2x - 12) is equal to -10. Remember, the absolute value of -10 is also 10. So, our second equation is: 2x - 12 = -10.

By creating these two equations, we've transformed one absolute value equation into two simpler linear equations. This is the magic trick to solving these problems. We've effectively removed the absolute value bars and created two paths to find the possible values of x.

Solving the First Equation: 2x - 12 = 10

Let's solve the first equation, 2x - 12 = 10. This is a straightforward linear equation, and we'll use basic algebraic principles to isolate x. Our goal is to get x by itself on one side of the equation.

First, we need to get rid of the -12. We can do this by adding 12 to both sides of the equation. Remember, whatever we do to one side, we must do to the other to keep the equation balanced. So, we have:

2x - 12 + 12 = 10 + 12

This simplifies to:

2x = 22

Now, we need to get rid of the 2 that's multiplying x. To do this, we'll divide both sides of the equation by 2:

2x / 2 = 22 / 2

This gives us our first solution:

x = 11

So, one possible value for x is 11. But remember, we have a second equation to solve, so we're not done yet! Stay focused, we're making great progress!

Solving the Second Equation: 2x - 12 = -10

Now, let's tackle the second equation: 2x - 12 = -10. We'll use the same steps as before to isolate x.

First, we add 12 to both sides of the equation:

2x - 12 + 12 = -10 + 12

This simplifies to:

2x = 2

Next, we divide both sides of the equation by 2:

2x / 2 = 2 / 2

This gives us our second solution:

x = 1

So, our second possible value for x is 1. We've now found two potential solutions for x: 11 and 1.

Checking the Solutions

It's always a good idea to check our solutions to make sure they're correct. This is especially important with absolute value equations, as we want to ensure that both values actually satisfy the original equation.

Let's start by checking x = 11. We'll substitute 11 for x in the original equation:

10 = |2(11) - 12|

10 = |22 - 12|

10 = |10|

10 = 10

This is true, so x = 11 is indeed a solution.

Now, let's check x = 1:

10 = |2(1) - 12|

10 = |2 - 12|

10 = |-10|

10 = 10

This is also true, so x = 1 is also a solution. Awesome! Both solutions check out.

Expressing the Solution in Simplest Form

We've found two values for x that satisfy the equation: x = 11 and x = 1. These are already in their simplest form, as they are integers. Therefore, our final solution is simply listing these two values.

We can express the solution set as {1, 11}. This notation indicates that the solution to the equation consists of these two numbers.

Conclusion

So, there you have it! We've successfully solved the absolute value equation 10 = |2x - 12|. Remember, the key to solving these types of equations is to recognize that the absolute value creates two possibilities, leading to two separate equations. By setting up and solving these equations, and then checking our answers, we can confidently find all the solutions.

Key takeaways from this problem:

• Absolute Value: Understand that absolute value represents distance from zero, which is always non-negative.
• Two Equations: Recognize that an absolute value equation generally leads to two separate equations.
• Solve and Check: Solve each equation individually and always check your solutions in the original equation.

I hope this explanation was helpful! Keep practicing, and you'll become a pro at solving absolute value equations in no time. Now you guys are equipped to tackle similar problems with confidence. Keep up the great work, and happy solving!