Solving the Equation 1/x - 1 4a with Absolute Values
Introduction to Absolute Algebraic Equations
Solving algebraic equations, especially those involving absolute values, can seem daunting at first. However, once you understand the fundamental principles, such equations become much more manageable. The equation in question, 1/x - 1 4a, can be challenging but is solvable with a step-by-step approach. This article will guide you through the process and explore various cases to find a solution.Understanding Absolute Algebraic Equations
An absolute algebraic equation is one where the variable is inside the absolute value function, often denoted as |f(x)|. The key principle to remember is that the absolute value of a number is always non-negative, meaning ( |f(x)| k ) results in two possible equations: 1. ( f(x) k ) 2. ( f(x) -k ) These two equations yield the complete set of solutions for the original absolute value equation. Let's apply this principle to our equation, 1/x - 1 4a.Case Analysis and Solving the Equation
Given the equation 1/x - 1 4a, we can consider multiple cases based on the value of 4a.Case 1: 4a ≥ 0
If 4a is non-negative, then we can solve directly. The equation simplifies as follows: [frac{1}{x-1} pm 4a] This gives us two sub-cases to consider:Sub-case 1.1: (frac{1}{x-1} 4a)
Solving for (x), we get: [frac{1}{x-1} 4a implies 1 4a(x-1) implies x-1 frac{1}{4a} implies x 1 frac{1}{4a} frac{4a 1}{4a}]Sub-case 1.2: (frac{1}{x-1} -4a)
Solving for (x) in this case gives: [frac{1}{x-1} -4a implies 1 -4a(x-1) implies x-1 -frac{1}{4a} implies x 1 - frac{1}{4a} frac{4a-1}{4a}] Therefore, the solutions for the equation 1/x - 1 4a when 4a ≥ 0 are: [boxed{ x frac{4a 1}{4a} text{ or } x frac{4a-1}{4a} }]Case 2: 4a If 4a is negative, the equation does not have any real solutions because the left-hand side of the original equation is always positive (since ( frac{1}{x-1} ) cannot be negative when subtracted from 1). Thus: [text{No solution when } 4a Special Cases and Clarifications The original equation you provided is ambiguous due to the lack of parentheses. Let's break it down into two options for clarity:
Option A: (left|frac{1}{x} - 1right| 4a)
Option B: (frac{1}{x} - 1 4a)
Let's solve these options step by step.Option A: (left|frac{1}{x} - 1right| 4a)
This equation involves absolute values. We need to consider two cases for the expression inside the absolute value:Case A.1: (frac{1}{x} - 1 4a)
Solving this, we get: [frac{1}{x} - 1 4a implies frac{1}{x} 4a 1 implies x frac{1}{4a 1}]Case A.2: (frac{1}{x} - 1 -4a)
Solving this, we get: [frac{1}{x} - 1 -4a implies frac{1}{x} 1 - 4a implies x frac{1}{1-4a}] Therefore, the solutions for Option A are: [boxed{ x frac{1}{4a 1} text{ or } x frac{1}{1-4a} }]