Solving Absolute Value Equations A Step By Step Guide To |4x 2|=10
Absolute value equations might seem daunting at first, but they become quite manageable once you understand the fundamental principles involved. This comprehensive guide will walk you through the step-by-step process of solving the absolute value equation |4x-2|=10. We'll break down the concept of absolute value, explore the different cases that arise, and provide clear, detailed explanations to ensure you grasp the method thoroughly. Whether you're a student grappling with algebra or simply looking to brush up on your math skills, this article will provide you with the knowledge and confidence to tackle similar problems.
Understanding Absolute Value
To effectively solve absolute value equations, it is crucial to first understand what absolute value represents. The absolute value of a number is its distance from zero on the number line. Distance is always non-negative, which means the absolute value of a number is always either positive or zero. For example, the absolute value of 5, denoted as |5|, is 5 because 5 is 5 units away from zero. Similarly, the absolute value of -5, denoted as |-5|, is also 5 because -5 is 5 units away from zero. This concept is the cornerstone for solving equations involving absolute values.
The absolute value function essentially transforms any negative number into its positive counterpart while leaving positive numbers unchanged. Mathematically, we can define the absolute value of a number x as follows:
|x| = x, if x ≥ 0
|x| = -x, if x < 0
This definition highlights the two critical cases we need to consider when solving absolute value equations. The expression inside the absolute value bars can be either positive or negative, and we must account for both possibilities to find all possible solutions. This understanding forms the basis for our approach in solving the equation |4x-2|=10.
When you encounter an absolute value equation, remember that you are essentially looking for all the numbers that, when plugged into the expression inside the absolute value bars, result in a distance from zero equal to the value on the other side of the equation. This often leads to two distinct equations to solve, one for the positive case and one for the negative case. Keeping this principle in mind will greatly simplify your approach to solving these types of equations and prevent common errors.
Setting Up the Two Cases for |4x-2|=10
Now that we have a strong grasp of absolute value, we can apply this knowledge to the equation |4x-2|=10. The core principle here is that the expression inside the absolute value bars, which is 4x-2, can be either 10 or -10. This is because both 10 and -10 have an absolute value of 10. Therefore, to solve this equation, we need to consider two separate cases:
Case 1: The Expression Inside is Positive
In this case, we assume that the expression 4x-2 is equal to 10. This gives us the equation:
4x - 2 = 10
This is a linear equation that we can solve using standard algebraic techniques. We will add 2 to both sides of the equation to isolate the term with x:
4x - 2 + 2 = 10 + 2
4x = 12
Next, we divide both sides by 4 to solve for x:
4x / 4 = 12 / 4
x = 3
So, the first potential solution is x = 3. This means that when x is 3, the expression 4x-2 equals 10, and the absolute value of 10 is indeed 10.
Case 2: The Expression Inside is Negative
The second case we need to consider is when the expression 4x-2 is equal to -10. This is because the absolute value of -10 is also 10. This gives us the equation:
4x - 2 = -10
Again, this is a linear equation that we can solve algebraically. We add 2 to both sides of the equation:
4x - 2 + 2 = -10 + 2
4x = -8
Now, we divide both sides by 4 to isolate x:
4x / 4 = -8 / 4
x = -2
Therefore, the second potential solution is x = -2. This means that when x is -2, the expression 4x-2 equals -10, and the absolute value of -10 is 10, satisfying the original equation.
By setting up and solving these two cases, we have accounted for both possibilities that arise from the absolute value. This methodical approach ensures that we find all possible solutions to the equation.
Solving Case 1: 4x - 2 = 10
As established earlier, in solving absolute value equations, the first case we consider for |4x-2|=10 is when the expression inside the absolute value, 4x-2, is equal to the positive value on the other side of the equation, which is 10. This gives us the linear equation:
4x - 2 = 10
The objective here is to isolate the variable x and determine its value. We achieve this by performing a series of algebraic operations on both sides of the equation, ensuring that the equation remains balanced. The first step is to eliminate the constant term (-2) on the left side. We do this by adding 2 to both sides of the equation:
4x - 2 + 2 = 10 + 2
Simplifying both sides, we get:
4x = 12
Now, we have the term 4x on the left side. To isolate x, we need to eliminate the coefficient 4. We accomplish this by dividing both sides of the equation by 4:
4x / 4 = 12 / 4
Performing the division, we find:
x = 3
Thus, the solution for Case 1 is x = 3. This means that if we substitute x = 3 back into the original absolute value equation, we should find that it holds true. Let's verify this:
|4(3) - 2| = |12 - 2| = |10| = 10
Since the result is 10, which is the value on the right side of the original equation, we have confirmed that x = 3 is indeed a valid solution. This thorough verification step is crucial to ensure accuracy, especially when dealing with absolute value equations where extraneous solutions can sometimes arise.
This methodical approach of adding or subtracting constants and then dividing by coefficients is fundamental in solving linear equations. By applying these steps systematically, you can confidently solve a wide range of algebraic problems. In the next section, we will tackle the second case for our absolute value equation, which involves considering the negative value.
Solving Case 2: 4x - 2 = -10
The second crucial step in solving the absolute value equation |4x-2|=10 involves considering the possibility that the expression inside the absolute value bars, 4x-2, might be equal to the negative of the value on the right side of the equation. In this case, that means we need to solve for when 4x-2 equals -10. This gives us the equation:
4x - 2 = -10
Just like in Case 1, our goal is to isolate the variable x. We will use the same algebraic principles to achieve this. First, we need to eliminate the constant term (-2) from the left side of the equation. To do this, we add 2 to both sides:
4x - 2 + 2 = -10 + 2
Simplifying both sides, we get:
4x = -8
Now, we have the term 4x on the left side. To isolate x completely, we must remove the coefficient 4. We do this by dividing both sides of the equation by 4:
4x / 4 = -8 / 4
Performing the division, we arrive at:
x = -2
So, the solution for Case 2 is x = -2. To ensure that this solution is valid, we must substitute x = -2 back into the original absolute value equation and check if it holds true:
|4(-2) - 2| = |-8 - 2| = |-10| = 10
As the result is 10, which matches the value on the right side of the original equation, we can confirm that x = -2 is indeed a valid solution. This step of verification is extremely important, as it helps us catch any potential errors and ensures that our solutions are accurate.
By solving both Case 1 and Case 2, we have explored all possibilities arising from the absolute value nature of the equation. The solutions we found, x = 3 and x = -2, represent the complete solution set for the equation |4x-2|=10. Understanding and applying this methodical approach will equip you to solve a wide variety of absolute value equations.
Verifying the Solutions
In solving absolute value equations, the final and arguably most critical step is verifying the solutions. This process ensures that the values we obtained for x are indeed correct and satisfy the original equation. Verification is especially crucial in absolute value equations because the nature of absolute values can sometimes lead to extraneous solutions—values that emerge during the solving process but do not actually satisfy the original equation. We have found two potential solutions for the equation |4x-2|=10: x = 3 and x = -2. To verify these solutions, we will substitute each value back into the original equation and check if the equation holds true.
Verifying x = 3
To verify x = 3, we substitute this value into the equation |4x-2|=10:
|4(3) - 2| = 10
First, we perform the multiplication inside the absolute value:
|12 - 2| = 10
Next, we simplify the expression inside the absolute value:
|10| = 10
The absolute value of 10 is 10, so we have:
10 = 10
Since the equation holds true, x = 3 is a valid solution.
Verifying x = -2
Now, we verify the second potential solution, x = -2, by substituting it into the original equation:
|4(-2) - 2| = 10
Again, we start by performing the multiplication inside the absolute value:
|-8 - 2| = 10
Then, we simplify the expression inside the absolute value:
|-10| = 10
The absolute value of -10 is 10, so we have:
10 = 10
Since this equation also holds true, x = -2 is also a valid solution.
By verifying both solutions, we have confirmed that both x = 3 and x = -2 are correct. This comprehensive verification process solidifies our understanding of the solution and eliminates any doubt about its accuracy. In the context of absolute value equations, this step is not just a formality but an essential part of the problem-solving process. It ensures that we provide the correct solutions and avoid any extraneous values that might arise.
Final Solutions and Conclusion
In conclusion, we have successfully solved the absolute value equation |4x-2|=10 by systematically breaking it down into two cases, solving each case separately, and then verifying the solutions. We first established the understanding of absolute value as the distance from zero, which led us to recognize that the expression inside the absolute value bars could be either positive or negative. This core principle is fundamental in tackling any absolute value equation.
We then set up two distinct cases:
- Case 1: 4x - 2 = 10, which we solved to find x = 3.
- Case 2: 4x - 2 = -10, which we solved to find x = -2.
After finding these potential solutions, we emphasized the critical step of verifying each solution by substituting it back into the original equation. This step is vital for ensuring accuracy and preventing extraneous solutions. We confirmed that both x = 3 and x = -2 satisfy the original equation |4x-2|=10.
Therefore, the final solutions to the equation |4x-2|=10 are:
- x = 3
- x = -2
This problem demonstrates a fundamental approach to solving absolute value equations: consider both the positive and negative cases, solve each case independently, and always verify your solutions. By following these steps, you can confidently solve a wide variety of absolute value equations. The key takeaway is the methodical approach—breaking down the problem into manageable parts and paying close attention to detail. This not only ensures accuracy but also deepens your understanding of the underlying mathematical principles.
Mastering the technique of solving absolute value equations is a valuable skill in algebra and beyond. It reinforces your understanding of equations, absolute value, and the importance of verification in mathematical problem-solving. With practice and a clear understanding of the steps involved, you can confidently tackle more complex equations and mathematical challenges.
