Problem 47 - College Algebra Essentials

Question

Contain rational equations with variables in denominators. For each equation, a. write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind , solve the equation. \(\frac{2}{x+1}-\frac{1}{x-1}=\frac{2 x}{x^{2}-1}\)

Step-by-Step Solution

Answer

The solution to the equation is \(x=0\). Restrictions were \(x=\pm1\), which don't affect our solution.

Step 1: Identifying Restrictions

Set each denominator equal to zero and solve for x: \n For \(x+1=0\), solution is \(x=-1\). This means -1 is a restriction. \n For \(x-1=0\), solution is \(x=1\). This is a restriction. \n For \(x^{2}-1=0\) solution is \(x=\pm1\). Both -1 and 1 are restrictions.

Step 2: Multiply through by the common denominator

Identify the common denominator which is \(x^{2}-1\) and multiply the whole equation by it to eliminate the fractions. This results in the equation \[ 2(x-1)-1(x+1)=2x \]. Simplifying it gives us \(x=0\).

Step 3: Check Restrictions

The last step is to check that the solution, \(x=0\), isn't a restriction. Our restrictions were \(x=\pm1\) which is not equal to \(x=0\). Hence, \(x=0\) is the solution to the equation.

Key Concepts

Restrictions

When solving rational equations, it's crucial to identify restrictions before proceeding to the solution. **Restrictions** are the values of variables that make any denominator equal zero, causing the expression to become undefined. For the equation \( \frac{2}{x+1}-\frac{1}{x-1}=\frac{2x}{x^{2}-1} \), these restrictions are found by setting each denominator to zero:

• For \( x+1=0 \), we have \( x=-1 \). Thus, \( x=-1 \) is a restriction.
• For \( x-1=0 \), we obtain \( x=1 \). Therefore, \( x=1 \) is another restriction.
• The denominator \( x^2-1 \) factors into \( (x-1)(x+1) \), and setting it to zero gives \( x=\pm1 \). Both these values were already identified as restrictions.

Understanding these restrictions is imperative because if any proposed solution coincides with a restriction, it cannot be considered valid. Always ensure that the final solution does not violate these restrictions before accepting it as a legitimate solution.

Common Denominator

One effective method for solving equations with rational expressions is to use a **common denominator**. This approach simplifies the equation by eliminating fractions. In our example, the expression \( \frac{2}{x+1}-\frac{1}{x-1}=\frac{2x}{x^{2}-1} \) requires identifying a common denominator.The common denominator is the least common multiple of all denominators involved. Here, it is \( x^2-1 \), which factors into \( (x+1)(x-1) \). By multiplying every term in the equation by \( x^2-1 \), the equation transforms to:

• The left side becomes \( 2(x-1) - 1(x+1) \)
• The fractions disappear, leaving \( 2x \) on the right side

This simplification steps ensure the variable terms are easier to deal with, allowing straightforward algebraic manipulation to solve for \( x \). Using the common denominator is a powerful technique in rational equations to streamline the solving process.

Variables in Denominators

Rational equations often include **variables in denominators**, like in \( \frac{2}{x+1} \) and \( \frac{1}{x-1} \). These variables mean that the value of the denominator varies with the variable, which is why identifying restrictions is so important.Variables in the denominators can lead to undefined expressions if the denominator equals zero. In our exercise, \( x \) is in each denominator expression:

• \( x+1 \), \( x-1 \), and \( x^2-1 \)

The presence of these variables requires setting up restrictions as discussed earlier to prevent undefined terms. When working with equations involving such variables, it is crucial to ensure that your calculations respect these constraints throughout the solution process.Overall, by remembering to first determine restrictions, employ a common denominator for simplification, and be mindful of variables in denominators, you can confidently tackle rational equations and find valid solutions.