Solving Absolute Value Equations A Step By Step Guide To |x 4|+6=13

Jul 5, 2025 by Admin 68 views

In the realm of mathematics, particularly when dealing with absolute value equations, a systematic approach is crucial to arrive at the correct solutions. Absolute value, denoted by vertical bars | |, represents the distance of a number from zero on the number line. This inherent property introduces the possibility of two scenarios: the expression inside the absolute value can be either positive or negative, both resulting in the same absolute value. In this comprehensive guide, we will dissect the equation $|x-4|+6=13$ , providing a step-by-step solution and elucidating the underlying principles of solving absolute value equations.

Understanding Absolute Value

Before diving into the solution, let's solidify our understanding of absolute value. The absolute value of a number, say 'a', is defined as follows:

|a| = a, if a ≥ 0 |a| = -a, if a < 0

This definition implies that the absolute value of a positive number or zero is the number itself, while the absolute value of a negative number is its opposite (positive) value. For instance, |5| = 5 and |-5| = -(-5) = 5. This concept is fundamental to solving absolute value equations, as we need to consider both positive and negative possibilities for the expression within the absolute value bars.

Isolating the Absolute Value

The first step in solving any absolute value equation is to isolate the absolute value expression. This means manipulating the equation to get the absolute value term alone on one side. In our case, the equation is $|x-4|+6=13$ . To isolate the absolute value, we subtract 6 from both sides of the equation:

$|x-4|+6-6=13-6$ $|x-4|=7$

Now, we have successfully isolated the absolute value expression, setting the stage for the next crucial step: considering both positive and negative cases.

Considering Positive and Negative Cases

With the absolute value isolated, we leverage the definition of absolute value to create two separate equations. The expression inside the absolute value, $(x-4)$ , can be either equal to the positive value on the right side (7) or the negative value (-7). This leads us to the following two cases:

Case 1: The Expression is Positive

If $(x-4)$ is positive or zero, then the absolute value $|x-4|$ is simply equal to $(x-4)$ . Thus, we have the equation:

$x-4=7$

Case 2: The Expression is Negative

If $(x-4)$ is negative, then the absolute value $|x-4|$ is equal to the negation of $(x-4)$ , which is $-(x-4)$ . This gives us the equation:

$-(x-4)=7$

Now, we have two linear equations that we can solve independently to find the possible values of x.

Solving the Linear Equations

Let's solve each of the linear equations we derived in the previous step:

Solving Case 1

The equation for Case 1 is $x-4=7$ . To solve for x, we add 4 to both sides:

$x-4+4=7+4$ $x=11$

So, one possible solution is $x=11$ .

Solving Case 2

The equation for Case 2 is $-(x-4)=7$ . To solve this, we first distribute the negative sign:

$-x+4=7$

Next, we subtract 4 from both sides:

$-x+4-4=7-4$ $-x=3$

Finally, we multiply both sides by -1 to solve for x:

$(-1)(-x)=(-1)(3)$ $x=-3$

Therefore, the second possible solution is $x=-3$ .

Verifying the Solutions

It's essential to verify our solutions by plugging them back into the original equation to ensure they are valid. This step is particularly crucial when dealing with absolute value equations, as extraneous solutions can sometimes arise.

Checking x = 11

Substitute $x=11$ into the original equation $|x-4|+6=13$ :

$|11-4|+6=13$ $|7|+6=13$ $7+6=13$ $13=13$

The equation holds true, so $x=11$ is a valid solution.

Checking x = -3

Substitute $x=-3$ into the original equation $|x-4|+6=13$ :

$|-3-4|+6=13$ $|-7|+6=13$ $7+6=13$ $13=13$

The equation holds true, confirming that $x=-3$ is also a valid solution.

Conclusion

In conclusion, the solutions to the absolute value equation $|x-4|+6=13$ are $x=11$ and $x=-3$ . This comprehensive guide has walked you through the process of solving absolute value equations, emphasizing the importance of isolating the absolute value, considering both positive and negative cases, solving the resulting linear equations, and verifying the solutions. By mastering these steps, you can confidently tackle a wide range of absolute value equations and deepen your understanding of mathematical concepts.

The correct answer is D. $x=11$ and $x=-3$ .

Now, let's analyze the given options and understand why option D is the correct answer.

A. $x=11$ and $x=-11$

We found that $x=11$ is a solution, but $x=-11$ is not. Let's substitute $x=-11$ into the original equation:

$|-11-4|+6=13$ $|-15|+6=13$ $15+6=13$ $21=13$

This is not true, so $x=-11$ is not a solution. Therefore, option A is incorrect.

B. $x=-11$ and $x=-3$

As we've already established, $x=-11$ is not a solution. While $x=-3$ is a solution, this option includes an incorrect solution. Thus, option B is incorrect.

C. $x=-11$ and $x=3$

Again, $x=-11$ is not a solution. Let's check $x=3$ :

$|3-4|+6=13$ $|-1|+6=13$ $1+6=13$ $7=13$

This is also not true, so $x=3$ is not a solution. Option C is incorrect.

D. $x=11$ and $x=-3$

We have already verified that both $x=11$ and $x=-3$ are solutions to the equation. Therefore, option D is the correct answer.

To further enhance your skills in solving absolute value equations, consider these helpful tips:

Isolate the Absolute Value: Always begin by isolating the absolute value expression. This simplifies the process and sets the stage for considering the two cases.
Consider Both Positive and Negative Cases: Remember that the expression inside the absolute value can be either positive or negative. This leads to two separate equations that need to be solved.
Solve the Linear Equations: After setting up the two cases, solve the resulting linear equations independently. Use standard algebraic techniques to find the possible values of the variable.
Verify the Solutions: Always verify your solutions by substituting them back into the original equation. This crucial step helps identify and eliminate any extraneous solutions that may arise.
Practice Regularly: Consistent practice is key to mastering any mathematical concept. Work through a variety of absolute value equations to build your confidence and problem-solving skills.
Understand the Definition: A solid understanding of the definition of absolute value is fundamental to solving these types of equations. Remember that absolute value represents the distance from zero, which is always non-negative.
Be Mindful of Extraneous Solutions: Extraneous solutions can sometimes occur when dealing with absolute value equations. These are solutions that satisfy the transformed equations but not the original equation. Verification helps catch these.

By following these tips and practicing regularly, you can become proficient in solving absolute value equations and excel in your mathematics studies.

While solving absolute value equations might seem like an abstract mathematical exercise, the concept of absolute value has numerous real-world applications. Here are a few examples:

Distance Calculations: Absolute value is inherently linked to the concept of distance. Whether calculating the distance between two points on a number line or the distance traveled, absolute value ensures the result is always positive.
Error Analysis: In scientific and engineering fields, absolute value is used to quantify the difference between an observed value and an expected value. This is crucial in error analysis and determining the accuracy of measurements.
Tolerance in Manufacturing: In manufacturing, absolute value is used to specify tolerances, which define the acceptable range of variation for a dimension. For instance, a part might have a specified length of 10 cm with a tolerance of ±0.1 cm, meaning the actual length can range from 9.9 cm to 10.1 cm. The absolute value ensures that the deviation from the specified length is within the acceptable range.
Computer Programming: Absolute value functions are commonly used in computer programming for various purposes, such as calculating distances, normalizing data, and handling conditional statements.
Finance: In finance, absolute value can be used to calculate the absolute change in an investment's value, regardless of whether it's a gain or a loss.

These examples illustrate that absolute value is not just a theoretical concept but a practical tool with wide-ranging applications in various fields. Understanding absolute value equations and their solutions can provide valuable insights into real-world problems.

By delving into these practical applications and consistently honing your problem-solving skills, you'll gain a deeper appreciation for the versatility and importance of mathematics in our everyday lives. Remember, the journey of mathematical exploration is a continuous one, filled with opportunities for discovery and growth. Embrace the challenges, celebrate the successes, and never cease to seek a more profound understanding of the mathematical world around us.