Problem 34 - Precalculus : Building Concepts and Connections

Question

Solve the rational equation. Check your solutions. $$\frac{x-1}{3 x+3}-\frac{9}{x^{2}-1}=\frac{2}{x+1}$$

Step-by-Step Solution

Answer

The solution for the rational equation is \(x = 6\).

Step 1 - Simplify the equation

First, simplify the denominator of the first term. The denominator can be factorized as \(3(x+1)\). Now, the equation becomes \(\frac{x-1}{3(x+1)}-\frac{9}{(x+1)(x-1)}=\frac{2}{x+1}\)

Step 2 - Find a common denominator

The least common denominator (LCD) of the equation is \(3(x+1)(x-1)\). Multiply every term by this LCD to get rid of the denominators. Doing this leads to: \(3(x-1) - 27 = 6(x-1)\)

Step 3 - Simplify and solve

This is a simplified linear equation. Start by expanding and simplifying: \(3x - 3 - 27 = 6x - 6\). Simplify further to \(3x - 30 = 6x -6\). Solve for X to yield the result \(x = 6\)

Step 4 - Check the solution

Finally, we should check that our solution, \(x = 6\), is valid by substituting the value back into the original equation: \(\frac{6-1}{3*6+3}-\frac{9}{6^{2}-1}=\frac{2}{6+1}\). Since both sides yield \(1=1\), the solution \(x = 6\) is indeed correct.

Key Concepts

Understanding Rational Equations

Rational equations are algebraic expressions that involve ratios of polynomials. A typical example of a rational equation is \( \frac{P(x)}{Q(x)} = \frac{R(x)}{S(x)} \), where \( P(x), Q(x), R(x) \) and \( S(x) \) are polynomials and \( Q(x) \) and \( S(x) \) are not equal to zero. These equations are particularly challenging because they often require finding a common denominator to combine terms and proceed with the solution.

When you encounter a rational equation, your goal is to combine all the terms over a single denominator so you can effectively 'clear' the denominators and solve the remaining equation. Always remember to check your solutions as they may not satisfy the original equation due to potential restrictions on the domain—values that would make any denominator zero.

Finding a Common Denominator

A common denominator is necessary when dealing with fractions that have different denominators
. In the context of rational equations, we look for the least common denominator (LCD) to combine terms. This LCD is the least common multiple of the denominators in the equation.

Finding the LCD helps us to combine fractions and work with a single fraction or eliminate the fractions entirely, making equations much easier to solve. It's like finding a common language between different countries—once you have it, everyone can communicate clearly. In the given exercise, the common denominator was \( 3(x+1)(x-1) \) which helped streamline the rational equation into a simpler form.

Factorization - The Key to Simplification

Factorization is essential in mathematics, especially when working with rational equations. It is the process of breaking down a complex expression into a product of simpler factors.

The ability to recognize and factor polynomials allows us to simplify fractions within an equation before solving. This simplification may reveal a common denominator, as seen in the textbook exercise with \( x^2 - 1 \) becoming \( (x+1)(x-1)\). Factorization can turn a daunting rational equation into manageable pieces, much like a difficult problem becomes easier to tackle when you break it down into smaller, more attainable steps.

Solving Linear Equations

Linear equations are the simplest form of equations to solve and they occur frequently in algebra. A linear equation is one where the variable(s), such as \( x \), have an exponent of 1. The standard form is \( ax + b = c \) where \( a \) and \( b \) are constants,

Once you have simplified a rational equation completely, you may find yourself left with a linear equation to solve. The process typically involves combining like terms and isolating the variable to one side of the equation. Successfully solving linear equations is a fundamental skill in algebra, as it is frequently the final step in solving more complex problems, like rational equations.