Problem 45 - Introductory Algebra for College Students

Question

Solve each rational equation. $$\frac{2}{x+3}-\frac{2 x+3}{x-1}=\frac{6 x-5}{x^{2}+2 x-3}$$

Step-by-Step Solution

Answer

The valid solutions would depend on the final simplified equation obtained and upon the validity checks.

Step 1: Identify common denominator

In order to simplify a complex fraction, find a common denominator. In this case, the common denominator is \((x+3)(x-1)(x^2+2x-3)\). You can determine this by simply multiplying all the denominators together, because they're all distinct.

Step 2: Multiply both sides of the equation

Multiply every term in the equation by \( (x+3)(x-1)(x^2+2x-3) \), so as to eliminate the denominators.

Step 3: Simplify the equation

The simplified equation after carrying out the multiplication should be \[2(x-1)(x^2+2x-3)-(2x+3)(x^2+2x-3)-(x+3)(6x-5) = 0\]. Perform the multiplication and combine like terms to simplify the equation.

Step 4: Solve for the variable

Find the value of x after simplifying the equation. Remember to test your solutions back in the original equation to make sure they are valid solutions, as certain values might lead to division by zero in the original equation and hence aren't valid.

Key Concepts

Understanding Common Denominator

When dealing with rational equations, finding a common denominator is essential. A common denominator is a shared multiple of all the denominators in the equation. It allows you to eliminate the fractions by bringing all terms to a common base, which simplifies the equation. To find this, you multiply the denominators given in your equation.
In the textbook exercise, the original denominators were \(x+3\), \(x-1\), and \(x^2+2x-3\). Multiplying these together gives the common denominator as \((x+3)(x-1)(x^2+2x-3)\).
This step is crucial because it ensures you are working with an equivalent equation free of fractions. Once the common denominator is determined, each term can be multiplied by it to simplify the equation's form.

What are Complex Fractions?

Complex fractions can be intimidating, but they are just fractions within fractions. They often arise in rational equations, leading to seemingly complicated expressions.
To simplify these, one of the most effective methods is to use the least common denominator. This technique allows the removal of all nested fractions, usually making the equation more straightforward to handle.

• Multiply each part of the complex fraction by the least common denominator of all the smaller fractions involved.
• This cancels out the smaller denominators, effectively un-nesting the fraction.

Using the common denominator \((x+3)(x-1)(x^2+2x-3)\), as seen in the exercise, transforms the complex structure into a simpler equation without fractional terms, making it more manageable to solve.

Techniques for Variable Solving

Once you have simplified the equation, the next step is solving for the variable. Variables are often isolated by converting the equation step by step until the variable is alone on one side.
Here's how you can achieve this:

• Simplify the equation by performing necessary algebraic operations like distribution, combining like terms, and rearranging terms.
• Factor the expression wherever possible, as factoring can simplify solving polynomial equations.
• After simplifying, solve for the variable typically through isolation.

In the given exercise, after eliminating the fractions and combining terms, you have a polynomial equation of the form \(2(x-1)(x^2+2x-3)-(2x+3)(x^2+2x-3)-(x+3)(6x-5) = 0\).
Checking your solution against the original equation is crucial, as some values might not be valid if they cause division by zero in the initial setup. Always verify your solutions to ensure they satisfy the original equation.