Solving the Equation x1 - x-1 2: A Comprehensive Guide
In this article, we will delve into the step-by-step process of solving the equation x1 - x-1 2. We will explore the use of absolute values, piecewise functions, and graphical representation to fully understand the solution.
Introduction
When faced with the equation x1 - x-1 2, it is essential to first understand the behavior of the absolute value function. The absolute value function |x| gives the distance of x from 0 on the number line. This concept will help us solve this equation through two main cases.
Solving the Equation
To solve the equation x1 - x-1 2, we need to consider two cases based on the signs of x and x-1.
Case 1: x1 ≥ 0 and x-1 ≥ 0
In this case, we can directly use x1 and x-1 instead of their absolute values. Substituting these into the original equation, we get:
[[ x^1 - x^{-1} 2]Let's simplify this equation:
[[ x - frac{1}{x} 2]By multiplying both sides by x, we get:
[[ x^2 - 2x - 1 0]This is a quadratic equation. Solving for x, we find:
[[ x 1 ± sqrt{2}]Since we are considering the case where x ≥ 1, the only valid solution is:
[[ x 1 sqrt{2}]We can verify this solution by substituting it back into the original equation:
[[ (1 sqrt{2}) - frac{1}{(1 sqrt{2})} 2]This solution satisfies the equation, confirming that the solution in this case is all x ≥ 1.
Case 2: x1 -1
In this case, x1 -x1 and x-1 -x-1. Substituting these into the original equation, we get:
[[ -x^1 - (-x^{-1}) 2]By simplifying, we have:
[[ -x^1 x^{-1} 2]Let's multiply both sides by x:
[[ -x^2 2x - 1 0]By rearranging, we find:
[[ x^2 - 2x 1 0]This equation simplifies to:
[[ (x - 1)^2 0]Solving for x, we get:
[[ x 1]However, we need to revisit our initial condition x
Graphical Representation
To further illustrate the solution, let's define the function:
[[ f(x) x^1 - x^{-1}]For x ≤ 1, we have:
[[ f(x) 2x]For x ≥ 1, we have:
[[ f(x) 2]"
Testing the Solution
We can test the solution with some values to verify its accuracy.
[[ text{For } x 2: 2^1 - 2^{-1} 2 - frac{1}{2} 2 - 0.5 1.5] [[ text{For } x 5: 5^1 - 5^{-1} 5 - frac{1}{5} 5 - 0.2 4.8]These values do not satisfy the equation, but if we test x ≥ 1, we can see:
[[ text{For } x 2: 2^1 - 2^{-1} 2 - frac{1}{2} 2 - 0.5 1.5] [[ text{For } x 5: 5^1 - 5^{-1} 5 - frac{1}{5} 5 - 0.2 4.8]Conclusion
In conclusion, the only valid solution to the equation x1 - x-1 2 is x ≥ 1. This solution is derived by considering the cases of x1 and x-1, using absolute value functions, and verifying the solutions through graphical representation and algebraic simplification.