Solving Absolute Value Equations A Step-by-Step Guide For |(1/2)x + 5| = 13
Hey guys! Let's dive into solving absolute value equations, specifically the equation |(1/2)x + 5| = 13. Absolute value equations might seem a bit tricky at first, but don't worry, we'll break it down step by step so you can master them. This guide will walk you through the process, explain the underlying concepts, and provide you with a clear understanding of how to tackle such problems. So, grab your pen and paper, and let's get started!
Understanding Absolute Value
Before we jump into solving the equation, it's essential to understand what absolute value really means. Absolute value represents the distance of a number from zero on the number line. Because distance is always non-negative, the absolute value of a number is always positive or zero. For example, the absolute value of 5 (written as |5|) is 5, and the absolute value of -5 (written as |-5|) is also 5. Think of it as stripping away the negative sign, if there is one.
Why Absolute Value Equations Have Two Solutions
The key thing to remember when dealing with absolute value equations is that they often have two possible solutions. This is because there are two numbers that are the same distance away from zero. Consider our equation, |(1/2)x + 5| = 13. This equation is asking: "What values of x will make the expression (1/2)x + 5 equal to either 13 or -13?" Both 13 and -13 have an absolute value of 13, which is why we need to consider both possibilities when solving. This dual nature of absolute value is crucial to grasp, and it's the foundation for solving these types of equations accurately. Many students initially forget to consider both positive and negative cases, leading to incomplete solutions. So, always remember to split the absolute value equation into two separate equations to account for both scenarios.
The Absolute Value Rule
In essence, when you encounter an absolute value equation like |expression| = value, you need to set up two separate equations:
- expression = value
- expression = -value
This rule is the cornerstone of solving absolute value equations. It ensures that you account for both the positive and negative scenarios that satisfy the absolute value condition. By applying this rule consistently, you can systematically break down complex equations into simpler, manageable parts. Keep this rule at the forefront of your mind as we proceed through the steps of solving our example equation. It’s not just a trick; it’s a direct consequence of the definition of absolute value, making it a powerful tool in your mathematical toolkit.
Step-by-Step Solution for |(1/2)x + 5| = 13
Now that we've got the basics down, let's tackle our equation: |(1/2)x + 5| = 13. We'll go through each step meticulously to ensure you understand the process.
Step 1: Split the Equation
As we discussed, the first step is to split the absolute value equation into two separate equations. This is because the expression inside the absolute value can be either 13 or -13. So, we set up the following two equations:
- (1/2)x + 5 = 13
- (1/2)x + 5 = -13
This is a critical step! By splitting the equation, we’re acknowledging that there are two possible scenarios that satisfy the original absolute value equation. Forgetting this step is a common mistake, so make sure you always remember to create these two separate equations. Each of these equations is now a standard linear equation, which we can solve using basic algebraic techniques. This split transforms the problem from a potentially confusing absolute value equation into two straightforward linear equations, making the rest of the solution process much clearer.
Step 2: Solve the First Equation (1/2)x + 5 = 13
Let's solve the first equation: (1/2)x + 5 = 13. Our goal here is to isolate x on one side of the equation.
First, we subtract 5 from both sides of the equation to get rid of the +5 on the left side:
(1/2)x + 5 - 5 = 13 - 5
This simplifies to:
(1/2)x = 8
Next, to get x by itself, we need to get rid of the (1/2) coefficient. We can do this by multiplying both sides of the equation by 2 (since multiplying by 2 is the inverse operation of multiplying by 1/2):
2 * (1/2)x = 2 * 8
This simplifies to:
x = 16
So, the first solution for x is 16. We've successfully isolated x in this equation by using inverse operations. Each step we took brought us closer to finding the value of x that satisfies this specific case. Remember to perform the same operation on both sides of the equation to maintain balance and ensure the equality remains true. This principle of maintaining balance is fundamental to solving any algebraic equation.
Step 3: Solve the Second Equation (1/2)x + 5 = -13
Now, let's tackle the second equation: (1/2)x + 5 = -13. We follow the same steps as before to isolate x.
First, subtract 5 from both sides:
(1/2)x + 5 - 5 = -13 - 5
This simplifies to:
(1/2)x = -18
Next, multiply both sides by 2:
2 * (1/2)x = 2 * (-18)
This simplifies to:
x = -36
Therefore, the second solution for x is -36. Just like in the first equation, we used inverse operations to isolate x and find its value. This systematic approach is essential for solving linear equations effectively. By consistently applying these steps, you can confidently solve a wide range of algebraic problems. Keep practicing, and you’ll find these techniques becoming second nature.
Step 4: Check Your Solutions
It's always a good idea to check your solutions to make sure they're correct. To do this, we'll plug each value of x back into the original equation |(1/2)x + 5| = 13 and see if it holds true.
Checking x = 16
Substitute x = 16 into the original equation:
|(1/2)(16) + 5| = 13
Simplify:
|8 + 5| = 13
|13| = 13
13 = 13
This is true, so x = 16 is a valid solution.
Checking x = -36
Substitute x = -36 into the original equation:
|(1/2)(-36) + 5| = 13
Simplify:
|-18 + 5| = 13
|-13| = 13
13 = 13
This is also true, so x = -36 is a valid solution.
By checking our solutions, we’ve confirmed that both x = 16 and x = -36 satisfy the original equation. This step is crucial because it helps prevent errors and ensures the accuracy of your work. It's a small investment of time that pays off by boosting your confidence in your answer. Always make it a habit to check your solutions, especially in exams or when accuracy is paramount.
Final Answer
So, the solutions to the equation |(1/2)x + 5| = 13 are x = 16 and x = -36. We've successfully navigated through the steps, from understanding the absolute value concept to solving the equation and verifying our answers.
Tips for Solving Absolute Value Equations
Before we wrap up, here are a few extra tips to keep in mind when solving absolute value equations:
Always Split the Equation
As we've emphasized, the most critical step is to split the absolute value equation into two separate equations. This accounts for both the positive and negative possibilities of the expression inside the absolute value. Never skip this step, as it’s fundamental to finding all solutions.
Isolate the Absolute Value First
Before splitting the equation, make sure the absolute value expression is isolated on one side of the equation. For example, if you have an equation like |2x + 3| + 4 = 9, you should first subtract 4 from both sides to get |2x + 3| = 5 before splitting it into two equations. This isolation step simplifies the process and prevents confusion.
Check for Extraneous Solutions
Sometimes, when solving absolute value equations, you might end up with solutions that don't actually satisfy the original equation. These are called extraneous solutions. That's why it's crucial to always check your solutions by plugging them back into the original equation. This ensures you only include valid solutions in your final answer.
Practice Makes Perfect
The best way to get comfortable with solving absolute value equations is to practice! Work through various examples, and you'll start to recognize patterns and become more confident in your ability to solve them. The more you practice, the easier it will become to tackle complex problems.
Conclusion
Solving absolute value equations doesn't have to be intimidating. By understanding the concept of absolute value, following a systematic approach, and remembering these tips, you can confidently solve equations like |(1/2)x + 5| = 13 and many others. Remember to split the equation, isolate the absolute value, check your solutions, and practice regularly. Keep up the great work, and you'll master these equations in no time! You've got this!
