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Magazine Chapter 2: Problem 18
Prove that if \(|f(x)|=|g(x)|\) then either \(f(x)=g(x)\) or \(f(x)=-g(x) .\) Use that result to solve the equations. \(|1-2 x|=|x+1|\)
Short Answer
Expert verified
The solutions are \(x = 0\) and \(x = 2\).
Step by step solution
01
Understand the Problem
We are given the equation \(|f(x)| = |g(x)|\) and need to prove that this results in either \(f(x) = g(x)\) or \(f(x) = -g(x)\). Once proven, we will use this to solve the equation \(|1 - 2x| = |x + 1|\).
02
Prove the Absolute Value Property
The absolute value of a number represents its distance from zero, which means for two numbers to have the same absolute value, they must either be equal or opposites. More formally, if \(|f(x)| = |g(x)|\), then either \(f(x) = g(x)\) or \(f(x) = -g(x)\). This results from the definition of absolute value: \(|a| = |b|\) implies \(a = b\) or \(a = -b\).
03
Apply the Proven Property to Given Equation
With \(|f(x)| = |g(x)|\) established as \(f(x) = g(x)\) or \(f(x) = -g(x)\), apply this to solve \(|1 - 2x| = |x + 1|\). This gives us two cases to consider: \((1 - 2x) = (x + 1)\) and \((1 - 2x) = -(x + 1)\).
04
Solve Case 1: \(1 - 2x = x + 1\)
Solve the equation \(1 - 2x = x + 1\). First, simplify the equation by subtracting \(x\) from both sides to get \(1 - 2x - x = 1\). This simplifies to \(1 - 3x = 1\). Now, subtract 1 from both sides to get \(-3x = 0\), and divide by -3 to find \(x = 0\).
05
Solve Case 2: \(1 - 2x = -(x + 1)\)
Solve the equation \(1 - 2x = -x - 1\). Distribute the negative on the right side, then simplify the equation by adding \(x\) to both sides to get \(1 - 2x + x = -1\). This simplifies to \(1 - x = -1\). Subtract 1 from both sides to have \(-x = -2\), and divide by -1 to find \(x = 2\).
06
Conclude the Solution
We solved the original equation \(|1 - 2x| = |x + 1|\) by considering two cases. The solutions are \(x = 0\) from Case 1 and \(x = 2\) from Case 2. Therefore, the complete solution to the equation consists of these two values.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Absolute Value Property
The absolute value of a number refers to its magnitude or distance from zero on the number line. This distance is always positive, which is why
- \(|a| = a\) if \(a \geq 0\), and
- \(|a| = -a\) if \(a < 0\).
- \(f(x) = g(x),\) or
- \(f(x) = -g(x)\).
Equivalent Equations
Equivalent equations are equations that have the same solutions. When we have the equation \(|f(x)| = |g(x)|\), applying the absolute value property lets us form two equivalent equations:
- \(f(x) = g(x)\) and
- \(f(x) = -g(x)\).
Case Analysis
Case analysis involves breaking down a problem into separate cases to simplify complex equations. In our context, it means using the absolute value property to consider the following scenarios:
- Case 1: \(f(x) = g(x)\)
- Case 2: \(f(x) = -g(x)\)
- Case 1 leads to solving: \(1 - 2x = x + 1\)
- Case 2 leads to solving: \(1 - 2x = -(x + 1)\)
Solving Equations
Solving equations involves finding all possible values for the variable that make the equation true. Here's how we solve each case for the equation \(|1 - 2x| = |x + 1|\):
- **Case 1:** Starting with \(1 - 2x = x + 1\), simplify to find \(1 - 3x = 1\), which leads us to \(x = 0\) after solving the linear equation.
- **Case 2:** For \(1 - 2x = -(x + 1)\), distribute the negative to get \(1 - 2x = -x - 1\). Simplifying, you find \(x = 2\) as a solution.
