Solving Absolute Value Equations Find Real Solutions For 4|x-5|=10

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In the realm of mathematics, absolute value equations often present a unique challenge, demanding a careful and systematic approach to unravel the potential solutions. This article delves into the intricacies of solving the absolute value equation 4|x-5|=10, providing a step-by-step guide to finding all real solutions. We'll explore the fundamental principles governing absolute values, the techniques for isolating the absolute value expression, and the methods for solving the resulting linear equations. This comprehensive exploration aims to equip readers with the knowledge and skills necessary to confidently tackle similar problems.

Understanding Absolute Value

Before diving into the solution, it's crucial to grasp the essence of absolute value. In mathematical terms, the absolute value of a number represents its distance from zero on the number line. It disregards the sign of the number, focusing solely on its magnitude. For instance, the absolute value of 5, denoted as |5|, is 5, and the absolute value of -5, denoted as |-5|, is also 5. This concept of distance forms the bedrock for solving absolute value equations.

The absolute value function can be formally defined as follows:

|x| = x, if x ≥ 0

|x| = -x, if x < 0

This definition highlights that the absolute value of a non-negative number is the number itself, while the absolute value of a negative number is its negation. This piecewise nature of the absolute value function is the key to understanding why absolute value equations often lead to multiple solutions. When dealing with absolute value equations, we must consider both scenarios: the expression inside the absolute value bars being positive or zero, and the expression inside the absolute value bars being negative.

Solving the Equation 4|x-5|=10

Now, let's embark on the journey of solving the equation 4|x-5|=10. Our objective is to isolate x, but the absolute value bars pose a hurdle. To overcome this, we'll employ a strategic approach involving isolating the absolute value expression and then splitting the equation into two separate linear equations.

Step 1: Isolate the Absolute Value Expression

The first step involves isolating the absolute value expression, |x-5|, on one side of the equation. To achieve this, we'll divide both sides of the equation by 4:

4|x-5| / 4 = 10 / 4

This simplifies to:

|x-5| = 5/2

Now, the absolute value expression is isolated, paving the way for the next crucial step.

Step 2: Split into Two Equations

The heart of solving absolute value equations lies in recognizing that the expression inside the absolute value bars can be either positive or negative. This realization leads us to split the equation into two separate linear equations:

  • Case 1: x-5 is positive or zero

    If x-5 ≥ 0, then |x-5| = x-5. The equation becomes:

    x - 5 = 5/2

  • Case 2: x-5 is negative

    If x-5 < 0, then |x-5| = -(x-5). The equation becomes:

    -(x - 5) = 5/2

We now have two linear equations, each representing a possible scenario for the value of x.

Step 3: Solve the Linear Equations

Let's solve each linear equation independently to find the potential solutions for x.

  • Solving Case 1: x - 5 = 5/2

    To isolate x, we add 5 to both sides of the equation:

    x - 5 + 5 = 5/2 + 5

    This simplifies to:

    x = 5/2 + 10/2

    x = 15/2

    Thus, one potential solution is x = 15/2.

  • Solving Case 2: -(x - 5) = 5/2

    First, we distribute the negative sign:

    -x + 5 = 5/2

    Next, we subtract 5 from both sides:

    -x + 5 - 5 = 5/2 - 5

    This simplifies to:

    -x = 5/2 - 10/2

    -x = -5/2

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

    x = 5/2

    Another potential solution is x = 5/2.

Step 4: Verify the Solutions

In the realm of absolute value equations, it's crucial to verify the solutions obtained. This step ensures that the solutions are valid and don't introduce any extraneous results. We'll substitute each potential solution back into the original equation to check its validity.

  • Verifying x = 15/2

    Substitute x = 15/2 into the original equation, 4|x-5|=10:

    4|15/2 - 5| = 10

    4|15/2 - 10/2| = 10

    4|5/2| = 10

    4 * (5/2) = 10

    10 = 10

    The equation holds true, confirming that x = 15/2 is a valid solution.

  • Verifying x = 5/2

    Substitute x = 5/2 into the original equation, 4|x-5|=10:

    4|5/2 - 5| = 10

    4|5/2 - 10/2| = 10

    4|-5/2| = 10

    4 * (5/2) = 10

    10 = 10

    The equation holds true, confirming that x = 5/2 is also a valid solution.

Final Solution

After meticulously solving and verifying the potential solutions, we arrive at the complete solution set for the equation 4|x-5|=10. The real solutions are:

x = 5/2, 15/2

These two values of x satisfy the original equation, representing the points where the absolute value expression fulfills the given condition.

Conclusion

Solving absolute value equations requires a methodical approach, encompassing the understanding of absolute value, the isolation of the absolute value expression, the splitting into linear equations, and the verification of solutions. By mastering these steps, you can confidently navigate the world of absolute value equations and unlock their solutions. The equation 4|x-5|=10 serves as a prime example, showcasing the techniques and considerations involved in finding real solutions. Remember to always verify your solutions to ensure accuracy and avoid extraneous results. With practice and a solid grasp of the underlying principles, absolute value equations will become less daunting and more approachable.

This article has provided a comprehensive guide to solving the absolute value equation 4|x-5|=10, equipping you with the knowledge and skills to tackle similar problems. As you delve deeper into the realm of mathematics, remember that practice and perseverance are key to mastering new concepts and techniques. Embrace the challenges, and you'll find yourself growing in mathematical prowess.