Problem 47 - College Algebra

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 given equation has no solution.

Step 1: Identify the Restrictions

To find the restrictions, set each denominator equal to zero and solve for \(x\). So, \(x+1=0\) gives \(x=-1\) and \(x-1=0\) gives \(x=1\). Also, for \(x^{2}-1=0\), solving gives \(x= -1\) or \(x=1\). Therefore, the restrictions for \(x\) are \(x=-1\) and \(x=1\). This means solution for \(x\) cannot be either -1 or 1.

Step 2: Obtain Common Denominator

The equation needs to be rewritten so that each term has the same denominator. In this case, the common denominator is \(x^2 - 1\), or \((x-1)(x+1)\). This gives: \[\frac{2}{x+1} \times \frac{x-1}{x-1} - \frac{1}{x-1} \times \frac{x+1}{x+1} - \frac{2x}{x^2-1}\] which simplifies to \[\frac{2(x-1)}{x^2-1} - \frac{x+1}{x^2-1} - \frac{2x}{x^2-1}\]

Step 3: Combine Terms & Solve

Combine the terms over the common denominator and then solve for \(x\): \[\frac{2x - 2 - x - 1 - 2x}{x^2-1} = 0\] This simplifies to \[\frac{-x-3}{x^2-1} = 0\] This equation gives -3 = 0, which is not correct. Hence there is no solution for this equation.

Key Concepts

Understanding Variables in Denominators

Rational equations frequently involve variables in the denominators, such as \( \frac{2}{x+1} \) and \( \frac{1}{x-1} \). When variables appear in the denominator, it's crucial to recognize that they create potential undefined expressions. This is because division by zero is undefined in mathematics. As a result, one of the first steps in solving rational equations is determining which values of the variable will make the denominator zero. Identifying these values early on allows us to avoid any mathematical mistakes or incorrect assumptions later in the problem-solving process.

In our original exercise, the denominators are \( x+1 \), \( x-1 \), and \( x^2-1 \). By setting each equal to zero, we determine that \( x = -1 \) and \( x = 1 \) will make the denominators zero. Knowing these values is important since they restrict the possible solutions of the equation.

Finding the Common Denominator

To solve rational equations efficiently, all terms should have a common denominator. This process simplifies the equation, making it easier to manipulate and eventually solve. The common denominator is typically the least common multiple of all individual denominators present in the equation.

In this exercise, we identified the common denominator as \( x^2 - 1 \), which can be factored as \((x-1)(x+1)\). Having a common denominator allows us to combine terms into a single fraction, which simplifies our equation significantly. Once combined, the equation can then be more readily solved for the variable of interest.

Considering Restrictions on Variables

Before solving any rational equation, it is vital to consider any restrictions on variables. Restrictions arise because having a zero in the denominator makes the equation undefined. Thus, values that lead to a zero denominator must be excluded from potential solutions.

In this situation, the restrictions were determined by setting the denominators \(x+1\), \(x-1\), and \(x^2-1\) equal to zero. By solving these simple equations, we found that \(x = -1\) and \(x = 1\) are the restricted values. Knowing these restrictions is essential, as they immediately indicate that any solution resulting in these values must be discarded.

• To avoid computational errors
• To ensure the solution is mathematically valid

Developing an intuitive understanding of these restrictions helps in ensuring that solutions are accurate.

Determining the Algebraic Solution

Once restrictions are understood and the equation is simplified with a common denominator, the next step is determining the algebraic solution. In this phase, all terms are combined over the common denominator, which allows the expression to be more easily solved for the variable.

In the exercise, combining the terms resulted in the equation \( \frac{-x-3}{x^2-1} = 0 \). To solve for \( x \), the equation must make sense algebraically. Here, the problem arises as the equation simplifies to \(-3 = 0\), which is clearly incorrect. This indicates that there is no solution for the given rational equation. It's helpful to realize that not all equations have solutions. Recognizing such results is an essential skill when dealing with rational equations.