Solving Absolute Value Equations: A Step-by-Step Guide

Hey guys! Today, we're diving into the world of absolute value equations. Specifically, we're going to break down how to solve the equation $-2|7-9 c|+7=-152+1-9 c-7$. Don't worry, it might look intimidating at first, but we'll tackle it together, step by step. By the end of this guide, you'll be a pro at solving these types of problems. So, grab your pencils, and let's get started!

Understanding Absolute Value

Before we jump into solving the equation, let's quickly recap what absolute value means. The absolute value of a number is its distance from zero on the number line. Because distance is always non-negative, the absolute value of a number is always positive or zero. We represent absolute value using vertical bars, like this: |x|. For example, |3| = 3 and |-3| = 3. Both 3 and -3 are three units away from zero.

Why is this important? Well, when we have an absolute value in an equation, it means we have to consider two possibilities: the expression inside the absolute value bars could be positive or negative. This is the key to solving absolute value equations correctly.

Breaking Down the Equation: $-2|7-9 c|+7=-152+1-9 c-7$

Now, let's get back to our equation: $-2|7-9 c|+7=-152+1-9 c-7$. The goal here is to isolate the variable 'c'. But first, we need to deal with that absolute value. Remember, it's like a package deal – we can't directly mess with what's inside the absolute value bars until we've isolated the absolute value term itself.

Step 1: Simplify and Isolate the Absolute Value Term

Our first task is to get the absolute value term, $-2|7-9c|$, by itself on one side of the equation. To do this, we'll perform a few algebraic operations. First, let’s simplify the right side of the equation by combining like terms:

$-152 + 1 - 9c - 7 = -158 - 9c$

So, our equation now looks like this:

$-2|7-9c| + 7 = -158 - 9c$

Next, we want to isolate the absolute value term. We can start by subtracting 7 from both sides of the equation:

$-2|7-9c| + 7 - 7 = -158 - 9c - 7$

This simplifies to:

$-2|7-9c| = -165 - 9c$

Now, we need to get rid of the -2 that's multiplying the absolute value. We can do this by dividing both sides of the equation by -2:

rac{-2|7-9c|}{-2} = rac{-165 - 9c}{-2}

This gives us:

|7-9c| = rac{165 + 9c}{2}

Great! We've successfully isolated the absolute value term. This is a crucial step because now we can address the two possibilities that the absolute value presents.

Step 2: Setting Up Two Equations

This is where the magic happens! Because the expression inside the absolute value, (7-9c), could be either positive or negative, we need to set up two separate equations:

Case 1: The expression inside the absolute value is positive or zero.

In this case, we simply remove the absolute value bars and keep the expression as is:

7 - 9c = rac{165 + 9c}{2}

Case 2: The expression inside the absolute value is negative.

In this case, we remove the absolute value bars and negate the expression inside:

-(7 - 9c) = rac{165 + 9c}{2}

Which simplifies to:

-7 + 9c = rac{165 + 9c}{2}

Now we have two equations, each representing a possible scenario. Let's solve them one at a time.

Step 3: Solving Case 1: 7 - 9c = rac{165 + 9c}{2}

To solve this equation, we first want to get rid of the fraction. We can do this by multiplying both sides of the equation by 2:

2(7 - 9c) = 2(rac{165 + 9c}{2})

This gives us:

$14 - 18c = 165 + 9c$

Next, we want to get all the 'c' terms on one side and the constants on the other. Let's add 18c to both sides:

$14 - 18c + 18c = 165 + 9c + 18c$

This simplifies to:

$14 = 165 + 27c$

Now, subtract 165 from both sides:

$14 - 165 = 165 + 27c - 165$

This gives us:

$-151 = 27c$

Finally, divide both sides by 27 to solve for 'c':

rac{-151}{27} = rac{27c}{27}

So, for Case 1, we have:

c = -rac{151}{27}

Step 4: Solving Case 2: -7 + 9c = rac{165 + 9c}{2}

We'll follow a similar process to solve this equation. First, multiply both sides by 2 to eliminate the fraction:

2(-7 + 9c) = 2(rac{165 + 9c}{2})

This simplifies to:

$-14 + 18c = 165 + 9c$

Next, subtract 9c from both sides:

$-14 + 18c - 9c = 165 + 9c - 9c$

This gives us:

$-14 + 9c = 165$

Now, add 14 to both sides:

$-14 + 9c + 14 = 165 + 14$

This simplifies to:

$9c = 179$

Finally, divide both sides by 9 to solve for 'c':

rac{9c}{9} = rac{179}{9}

So, for Case 2, we have:

c = rac{179}{9}

Step 5: Checking Your Solutions

It's super important to check our solutions to make sure they actually work in the original equation. Absolute value equations can sometimes have extraneous solutions (solutions that don't satisfy the original equation), so we need to be careful.

Checking c = -rac{151}{27}:

Plug this value back into the original equation: $-2|7-9 c|+7=-152+1-9 c-7$

-2|7 - 9(-rac{151}{27})| + 7 = -152 + 1 - 9(-rac{151}{27}) - 7

Simplify:

-2|7 + rac{151}{3}| + 7 = -158 + rac{151}{3}

-2|rac{21 + 151}{3}| + 7 = rac{-474 + 151}{3}

-2|rac{172}{3}| + 7 = rac{-323}{3}

-2(rac{172}{3}) + 7 = rac{-323}{3}

-rac{344}{3} + rac{21}{3} = rac{-323}{3}

rac{-323}{3} = rac{-323}{3}

This solution checks out!

Checking c = rac{179}{9}:

Plug this value back into the original equation:

-2|7 - 9(rac{179}{9})| + 7 = -152 + 1 - 9(rac{179}{9}) - 7

Simplify:

$-2|7 - 179| + 7 = -158 - 179$

$-2|-172| + 7 = -337$

$-2(172) + 7 = -337$

$-344 + 7 = -337$

$-337 = -337$

This solution also checks out!

Step 6: State Your Solution

We've done it! We've solved the equation and checked our solutions. The solutions to the equation $-2|7-9 c|+7=-152+1-9 c-7$ are:

c = -rac{151}{27} and c = rac{179}{9}

Key Takeaways for Solving Absolute Value Equations

Before we wrap up, let's quickly review the key steps for solving absolute value equations:

1. Isolate the absolute value term: Get the term with the absolute value by itself on one side of the equation.
2. Set up two equations: Create two separate equations by considering both the positive and negative cases of the expression inside the absolute value bars.
3. Solve each equation: Solve each equation independently using standard algebraic techniques.
4. Check your solutions: Plug each solution back into the original equation to make sure it works. This is crucial to avoid extraneous solutions.
5. State your solution: Clearly state the values of the variable that satisfy the equation.

Wrapping Up

And there you have it! Solving absolute value equations might seem tricky at first, but by breaking them down into manageable steps, you can conquer even the most complex problems. Remember to always isolate the absolute value, set up your two cases, and definitely check your answers. With practice, you'll become a master of absolute value equations. Keep up the great work, guys!