Solving Absolute Value Equations: |5x+2| = |7x+3|

Hey guys! Let's dive into solving an absolute value equation. Specifically, we're tackling the equation |5x + 2| = |7x + 3|. Absolute value equations might seem a bit intimidating at first, but don't worry; we'll break it down step by step so it’s super easy to understand. Stick around, and you'll be solving these like a pro in no time!

Understanding Absolute Value

Before we jump into the nitty-gritty of solving the equation, let's quickly recap what absolute value actually means. The absolute value of a number is its distance from zero on the number line. So, whether you're dealing with a positive number or a negative number, the absolute value is always non-negative.

For example:

• |5| = 5 because 5 is 5 units away from zero.
• |-5| = 5 because -5 is also 5 units away from zero.

Knowing this, we can understand that when we have an equation like |x| = a, it really means that x could be either a or -a. That’s because both a and -a are 'a' units away from zero. Keep this in mind as we move forward; it's super important for tackling our main equation.

Setting Up the Cases

Okay, now that we're all on the same page about absolute values, let's get back to our equation: |5x + 2| = |7x + 3|. Because we have absolute values on both sides, we need to consider a couple of different cases to make sure we cover all possible solutions.

The main idea here is that either the expressions inside the absolute value signs are equal to each other, or they are the negative of each other. This gives us two cases to consider:

Case 1: The expressions inside the absolute values are equal.

This means: 5x + 2 = 7x + 3

Case 2: The expressions inside the absolute values are the negative of each other.

This means: 5x + 2 = -(7x + 3)

By considering both of these cases, we ensure that we find all possible values of x that satisfy the original equation. Make sense? Great! Now, let’s solve each case separately.

Solving Case 1: 5x + 2 = 7x + 3

Alright, let's dive into solving the first case: 5x + 2 = 7x + 3. Our goal here is to isolate x on one side of the equation. To do that, we'll use some basic algebraic manipulation. Don't worry; it's easier than it sounds!

First, let’s subtract 5x from both sides of the equation. This will get all our x terms on one side:

5x + 2 - 5x = 7x + 3 - 5x

This simplifies to:

2 = 2x + 3

Next, we want to get the constant terms on the other side, so let’s subtract 3 from both sides:

2 - 3 = 2x + 3 - 3

Which simplifies to:

-1 = 2x

Now, to finally isolate x, we divide both sides by 2:

-1 / 2 = 2x / 2

This gives us:

x = -1/2

So, the solution for Case 1 is x = -1/2. Awesome, right? But hold on, we’re not done yet. We still need to solve Case 2 to see if there are any other possible solutions.

Solving Case 2: 5x + 2 = -(7x + 3)

Okay, time to tackle the second case: 5x + 2 = -(7x + 3). This one involves a negative sign, so we need to be extra careful with our distribution.

First, let's distribute that negative sign on the right side of the equation:

5x + 2 = -7x - 3

Now, we want to get all the x terms on one side. Let’s add 7x to both sides:

5x + 2 + 7x = -7x - 3 + 7x

This simplifies to:

12x + 2 = -3

Next, we want to isolate the x term, so let’s subtract 2 from both sides:

12x + 2 - 2 = -3 - 2

Which simplifies to:

12x = -5

Finally, let's divide both sides by 12 to solve for x:

12x / 12 = -5 / 12

This gives us:

x = -5/12

So, the solution for Case 2 is x = -5/12. Alright, we’ve got two potential solutions now. What’s next?

Checking the Solutions

Now that we've found two potential solutions, x = -1/2 and x = -5/12, we need to make sure they actually work. The best way to do this is to plug each value back into the original equation and see if it holds true.

Checking x = -1/2:

Let’s substitute x = -1/2 into the original equation |5x + 2| = |7x + 3|:

|5(-1/2) + 2| = |7(-1/2) + 3|

|-5/2 + 2| = |-7/2 + 3|

|-5/2 + 4/2| = |-7/2 + 6/2|

|-1/2| = |-1/2|

1/2 = 1/2

Since this is true, x = -1/2 is indeed a valid solution.

Checking x = -5/12:

Now let’s substitute x = -5/12 into the original equation:

|5(-5/12) + 2| = |7(-5/12) + 3|

|-25/12 + 2| = |-35/12 + 3|

|-25/12 + 24/12| = |-35/12 + 36/12|

|-1/12| = |1/12|

1/12 = 1/12

Since this is also true, x = -5/12 is a valid solution as well.

Final Answer

So, after solving both cases and checking our solutions, we've found that the solutions to the equation |5x + 2| = |7x + 3| are:

x = -1/2 and x = -5/12

That's it! You've successfully solved an absolute value equation. Give yourself a pat on the back! These problems might seem tricky at first, but with a little practice, you'll become a master at them. Keep up the great work, and happy solving!

Conclusion

Wrapping things up, solving absolute value equations involves considering different cases to account for the possible positive and negative values inside the absolute value signs. Remember to:

1. Set up the cases: Determine the possible scenarios based on the absolute values.
2. Solve each case: Use algebraic manipulation to find potential solutions.
3. Check the solutions: Plug the potential solutions back into the original equation to verify they are valid.

By following these steps, you can confidently solve a wide range of absolute value equations. Keep practicing, and you’ll find these problems become second nature! You got this!