Solving Absolute Value Equations Find The Values Of X In |-8x| = 72
In the realm of mathematics, absolute value equations often present a unique challenge that requires careful consideration of both positive and negative possibilities. This article delves into the process of solving the absolute value equation |-8x| = 72, providing a comprehensive guide to understanding the underlying principles and arriving at the correct solutions. Understanding absolute value is crucial not only for solving equations but also for grasping concepts in various branches of mathematics, including calculus and linear algebra.
Understanding Absolute Value
At its core, the absolute value of a number represents its distance from zero on the number line, irrespective of its sign. This distance is always a non-negative value. Mathematically, the absolute value of a number 'a' is denoted as |a| and 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 is the number itself, while the absolute value of a negative number is its positive counterpart. For instance, |5| = 5 and |-5| = 5. This fundamental concept is the cornerstone of solving absolute value equations.
Absolute value equations arise in numerous mathematical contexts, from basic algebra to more advanced topics. Mastering the techniques for solving them is therefore essential for any student of mathematics. The equation |-8x| = 72 is a classic example that allows us to illustrate these techniques clearly.
Solving the Equation |-8x| = 72
To solve the equation |-8x| = 72, we must recognize that the expression inside the absolute value, -8x, can be either 72 or -72, since both |72| and |-72| equal 72. This understanding leads us to split the original equation into two separate equations:
- -8x = 72
- -8x = -72
Each of these equations can be solved independently using basic algebraic principles. Solving them correctly is crucial to finding all possible values of x that satisfy the original equation.
Solving the First Equation: -8x = 72
To isolate x in the equation -8x = 72, we need to divide both sides of the equation by -8. This operation will undo the multiplication by -8, allowing us to determine the value of x.
Dividing both sides by -8, we get:
x = 72 / -8 x = -9
Therefore, the first solution to the equation |-8x| = 72 is x = -9. This value of x makes the expression inside the absolute value equal to 72, which satisfies the original equation.
Solving the Second Equation: -8x = -72
Similarly, to solve the second equation -8x = -72, we again need to isolate x by dividing both sides of the equation by -8. This step is essential to find the second possible value of x that satisfies the given absolute value equation.
Dividing both sides by -8, we obtain:
x = -72 / -8 x = 9
Thus, the second solution to the equation |-8x| = 72 is x = 9. This value of x makes the expression inside the absolute value equal to -72, whose absolute value is also 72, thereby satisfying the original equation.
Verifying the Solutions
To ensure the accuracy of our solutions, it is always a good practice to substitute the values we found back into the original equation. This verification step helps to confirm that our solutions are indeed correct and that no errors were made during the solving process.
Verifying x = -9
Substituting x = -9 into the original equation |-8x| = 72, we get:
|-8(-9)| = 72 |72| = 72 72 = 72
Since the equation holds true, we can confidently say that x = -9 is a valid solution.
Verifying x = 9
Substituting x = 9 into the original equation |-8x| = 72, we get:
|-8(9)| = 72 |-72| = 72 72 = 72
Again, the equation holds true, confirming that x = 9 is also a valid solution.
Conclusion: The Solutions to |-8x| = 72
In conclusion, by systematically solving the absolute value equation |-8x| = 72, we have identified two distinct solutions: x = -9 and x = 9. These solutions represent the values of x that, when substituted into the original equation, make the equation true. The process involved understanding the definition of absolute value, splitting the equation into two cases, solving each case independently, and verifying the solutions to ensure their correctness.
Understanding and solving absolute value equations is a fundamental skill in mathematics. It not only reinforces basic algebraic techniques but also lays the groundwork for more advanced mathematical concepts. The ability to solve such equations efficiently and accurately is invaluable in various fields, including engineering, physics, and computer science.
This article has provided a detailed explanation of how to solve the specific equation |-8x| = 72. The techniques and principles discussed can be applied to solve a wide range of absolute value equations, making this a valuable resource for students and anyone interested in mathematics. By mastering these skills, you can confidently tackle more complex mathematical problems and deepen your understanding of the mathematical world.
By understanding the concept of absolute value and applying the steps outlined in this article, you can confidently solve similar problems. Remember, the key is to consider both positive and negative possibilities for the expression inside the absolute value bars. Keep practicing, and you'll master these equations in no time!
