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Chapter 1: Problem 33

Find the real solutions, if any, of each equation. $$ \left|\frac{5 x-3}{3 x-5}\right|=2 $$

Short Answer

Expert verified
The real solutions are \(x = 7\) and \(x = \frac{13}{11}\).

Step by step solution

01

Set Up Two Possible Equations

Since the absolute value of an expression equals 2, we have two cases to consider: Case 1: \[ \frac{5x - 3}{3x - 5} = 2 \]Case 2: \[ \frac{5x - 3}{3x - 5} = -2 \]
02

Solve Case 1

For the first case, solve the equation \[ \frac{5x - 3}{3x - 5} = 2 \]Step 2a: Multiply both sides by \(3x - 5\) to eliminate the fraction: \[ 5x - 3 = 2(3x - 5) \]Step 2b: Distribute and simplify: \[ 5x - 3 = 6x - 10 \]Step 2c: Rearrange the equation to isolate \(x\): \[ 5x - 6x = -10 + 3 \]\[ -x = -7 \]\[ x = 7 \]
03

Solve Case 2

For the second case, solve the equation \[ \frac{5x - 3}{3x - 5} = -2 \]Step 3a: Multiply both sides by \(3x - 5\) to eliminate the fraction: \[ 5x - 3 = -2(3x - 5) \]Step 3b: Distribute and simplify: \[ 5x - 3 = -6x + 10 \]Step 3c: Rearrange the equation to isolate \(x\): \[ 5x + 6x = 10 + 3 \]\[ 11x = 13 \]\[ x = \frac{13}{11} \]
04

Check for Restrictions

Check if either solution makes the denominator zero, as denominators cannot be zero. For \(x = 7\): \[ 3(7) - 5 = 21 - 5 = 16 \] which is not zero. For \(x = \frac{13}{11}\): \[ 3 \left( \frac{13}{11} \right) - 5 = \frac{39}{11} - \frac{55}{11} = \frac{-16}{11} \] which is not zero. Both solutions are valid.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

solving equations
In algebra, solving equations involves finding the value of the variable that makes the equation true. When solving equations, it's essential to isolate the variable. This usually entails performing the same operation on both sides of the equation. In the given problem, we need to solve for the variable, which is represented by solving two cases due to the absolute value concept. Absolute value equations require you to consider both the positive and negative scenarios of the expression inside the absolute value. This means setting up two separate equations and solving each one independently.
rational expressions
Rational expressions are fractions where the numerator and the denominator are polynomials. A key aspect of working with rational expressions is to simplify them, sometimes by factoring polynomials and canceling common factors. In our problem, the term \(\frac{5x - 3}{3x - 5}\) is a rational expression. Another important step is to consider the domain of the expression, which is all real numbers except where the denominator equals zero, as division by zero is undefined. Solving rational expressions often involves cross-multiplication to eliminate the fraction form, which simplifies the equation and helps us find the solution.
algebraic fractions
Algebraic fractions are fractions where the numerator and/or the denominator contain algebraic expressions. When solving equations involving these fractions, like in our problem, adhering to specific steps is crucial. Starting with isolating the fraction, then multiplying through by the denominator to clear the fraction helps simplify the problem. Once the fraction is eliminated, the equation converts to a simpler linear or quadratic equation, which can be solved by usual algebraic methods. Remember to always check the solutions in the context of the original fraction to ensure they do not make the denominator zero, thereby invalidating the solutions.

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