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

Oct 28, 2025 by SLV Team 55 views

Hey guys! Today, we're diving into the world of absolute value equations and tackling a specific problem: $-3|2x+5|-4=-16$ . Don't worry, it might look intimidating at first, but we'll break it down step by step so it's super easy to understand. Absolute value equations might seem tricky, but they're totally manageable once you get the hang of the basic principles. We'll walk through each step, explaining the why behind the what, so you can confidently solve similar problems on your own. So, buckle up and let's get started!

Understanding Absolute Value

Before we jump into solving the equation, let's quickly recap what absolute value actually means. Absolute value essentially represents the distance of a number from zero on the number line. Because distance is always a non-negative value, the absolute value of a number is always positive or zero. For example, the absolute value of 5, written as |5|, is 5 because 5 is 5 units away from zero. Similarly, the absolute value of -5, written as |-5|, is also 5 because -5 is also 5 units away from zero. Understanding this concept is crucial for solving equations involving absolute values.

When dealing with absolute value equations, we need to remember that the expression inside the absolute value bars can be either positive or negative, and both possibilities need to be considered. This is the key to unlocking the solutions. We are not just looking for one answer, but rather all possible values of 'x' that satisfy the given equation. Keep this in mind as we move forward; it's the core principle that guides our approach. Absolute value is used extensively in various fields of mathematics and physics, so grasping this concept is beneficial beyond just solving this specific equation. We often see it in distance calculations, error analysis, and various types of inequalities. Therefore, a solid understanding here will certainly pay off in your future mathematical endeavors.

Isolating the Absolute Value Expression

Our first goal in solving the equation $-3|2x+5|-4=-16$ is to isolate the absolute value expression, which is $|2x+5|$ . Think of it like peeling an onion – we need to get rid of the layers surrounding the absolute value before we can deal with the expression inside it.

To do this, we'll use basic algebraic operations. First, we need to get rid of the -4 that's hanging out on the left side of the equation. We can do this by adding 4 to both sides of the equation. Remember, whatever we do to one side, we must do to the other to keep the equation balanced. This gives us: $-3|2x+5| = -12$ . Now, we're one step closer! Next, we need to get rid of the -3 that's multiplying the absolute value. To do this, we'll divide both sides of the equation by -3. This gives us: $|2x+5| = 4$ . Awesome! We've successfully isolated the absolute value expression. Now, we're ready to tackle the heart of the problem: understanding what this equation tells us. Remember, isolating the absolute value is a crucial step in solving these types of equations. It sets the stage for the next part, where we'll consider both the positive and negative possibilities of the expression inside the absolute value bars.

Considering Both Positive and Negative Cases

This is where the magic happens! Since the absolute value of an expression represents its distance from zero, there are two possibilities for the expression inside the absolute value bars to equal 4. The expression $(2x + 5)$ could be equal to 4, or it could be equal to -4. This is because both 4 and -4 have an absolute value of 4. So, we need to consider both scenarios separately.

This is the key concept in solving absolute value equations. We split the problem into two separate equations, each representing one of the possibilities. By doing this, we ensure that we don't miss any potential solutions. For Case 1, we assume the expression inside the absolute value is positive, so we have $2x + 5 = 4$ . For Case 2, we assume the expression inside the absolute value is negative, so we have $2x + 5 = -4$ . Now we have two simple linear equations that we can solve independently. This approach transforms a seemingly complex problem into two manageable ones. Remember, it’s essential to consider both possibilities to ensure you find all the solutions to the original absolute value equation. Let's solve each of these cases in the next section.

Solving for x in Each Case

Now that we've split the problem into two cases, let's solve for x in each one. This involves using basic algebraic manipulation to isolate x on one side of the equation.

Case 1: $2x + 5 = 4$ . To solve for x, we first subtract 5 from both sides, which gives us $2x = -1$ . Then, we divide both sides by 2, which gives us $x = - rac{1}{2}$ . So, one possible solution is $x = - rac{1}{2}$ .

Case 2: $2x + 5 = -4$ . Again, we start by subtracting 5 from both sides, which gives us $2x = -9$ . Then, we divide both sides by 2, which gives us $x = - rac{9}{2}$ . So, another possible solution is $x = - rac{9}{2}$ . We now have two potential solutions. It's a good practice to verify these solutions in the original equation to make sure they are valid. This process helps to catch any potential errors made during the solving process. Remember, accuracy is key in mathematics, so taking that extra step to verify your answers is always a good idea. Now, let's verify these solutions in the original equation.

Verifying the Solutions

It's always a good idea to verify our solutions to make sure they actually work in the original equation. This helps us catch any errors we might have made along the way. We'll plug each of our potential solutions, $x = - rac{1}{2}$ and $x = - rac{9}{2}$ , back into the original equation, $-3|2x+5|-4=-16$ , and see if they make the equation true.

Let's start with $x = - rac{1}{2}$ :

Plug it in: $-3|2(- rac{1}{2})+5|-4 = -3|-1+5|-4 = -3|4|-4 = -3(4)-4 = -12-4 = -16$ . Woohoo! It works! So, $x = - rac{1}{2}$ is indeed a valid solution.

Now, let's check $x = - rac{9}{2}$ :

Plug it in: $-3|2(- rac{9}{2})+5|-4 = -3|-9+5|-4 = -3|-4|-4 = -3(4)-4 = -12-4 = -16$ . Awesome! This one works too! So, $x = - rac{9}{2}$ is also a valid solution. Verifying our solutions confirms that we have accurately solved the equation. It's a crucial step in problem-solving to ensure that the answers we arrive at are correct. This practice also builds confidence in your problem-solving skills. So, we can confidently state that the solutions to the equation are $x = - rac{1}{2}$ and $x = - rac{9}{2}$ .

The Final Answer

After all that work, we've arrived at the final answer! We found two solutions for the equation $-3|2x+5|-4=-16$ . These solutions are $x = - rac{1}{2}$ and $x = - rac{9}{2}$ . Therefore, the correct answer is B. $x=- rac{1}{2}$ or $x=- rac{9}{2}$ .

Hopefully, this step-by-step guide has made solving absolute value equations a little less mysterious. Remember the key steps: isolate the absolute value expression, consider both positive and negative cases, solve for x in each case, and verify your solutions. With practice, you'll become a pro at solving these types of equations. Keep up the great work, and don't be afraid to tackle challenging problems! Remember, math can be fun, especially when you understand the concepts and the steps involved.

Tips for Solving Absolute Value Equations

To wrap things up, let’s go over some crucial tips that can help you tackle absolute value equations with confidence:

Isolate the Absolute Value: This is the golden rule. Before doing anything else, make sure the absolute value expression is by itself on one side of the equation. This might involve adding, subtracting, multiplying, or dividing both sides of the equation.
Consider Both Positive and Negative Cases: Remember that the expression inside the absolute value bars can be either positive or negative. Split the problem into two separate equations, one for each possibility.
Solve Each Case Independently: Once you have your two equations, solve each one for the variable. Use standard algebraic techniques for solving linear equations.
Verify Your Solutions: Always, always, always check your solutions by plugging them back into the original equation. This is essential to ensure you haven’t made any mistakes and that the solutions are valid.
Watch out for Extraneous Solutions: Sometimes, when solving absolute value equations, you might get solutions that don’t actually work in the original equation. These are called extraneous solutions. Verifying your solutions helps you identify and eliminate them.

By keeping these tips in mind, you’ll be well-equipped to solve a wide range of absolute value equations. Practice is key, so don't hesitate to work through more examples and solidify your understanding. You've got this! Remember, the more you practice, the more comfortable you'll become with these types of problems.