Problem 81 - Prealgebra and Introductory Algebra

Question

Simplify. $$\frac{3 x+1}{x-1}-\frac{x-1}{x-3}+\frac{x+1}{x^{2}-4 x+3}$$

Step-by-Step Solution

Answer

The simplified expression from the given complex fraction is \(\frac{x^2 +x-1.5}{(x-1)(x-3)}\).

Step 1: Factorization

Factor the denominator of the last term \((x^{2}-4 x+3)\) as \((x-1)(x-3)\). Therefore, the given expression becomes \(\frac{3x+1}{x-1}-\frac{x-1}{x-3}+\frac{x+1}{(x-1)(x-3)}\).

Step 2: Find a common denominator

In order to simplify this expression, a common denominator is required. In this case it is \((x-1)(x-3)\). Now transform the fraction: first fraction needs \((x-3)\) to match the common denominator and second fraction needs \((x-1)\) so we get: \(\frac{(3x+1)(x-3)}{(x-1)(x-3)} - \frac{(x-1)(x-1)}{(x-1)(x-3)} + \frac{x+1}{(x-1)(x-3)}\).

Step 3: Perform the multiplication in the numerators

Multilpy the numerators of first two fractions: \(\frac{(3x^2-8x-3)-(x^2-2x+1)+(x+1)}{(x-1)(x-3)}\).

Step 4: Simplify the numerator

Combine like terms in the numerator: \(\frac{3x^2-x^2-x+2x-3-1+x+1}{(x-1)(x-3)}\), which simplifies to: \(\frac{2x^2 +2x-3}{(x-1)(x-3)}\).

Step 5: Simplification

Finally, simplify the fraction by reducing the common factor of 2 in the numerator and denominator: \(\frac{x^2 +x-1.5}{(x-1)(x-3)}\).

Key Concepts

Factorization

Understanding how to simplify algebraic fractions begins with the concept of factorization. Factorization is the process of breaking down an expression into a product of its factors. Factors are expressions you multiply together to get the original expression. Especially in algebra, factorization helps simplify complex fractions and solve equations.
For instance, when you have a denominator like \(x^2 - 4x + 3\), you can factorize it into \(x-1\) and \(x-3\). Identifying these factors is crucial, as it enables you to combine fractions that might initially seem unrelated. Just like in the exercise, where the third fraction \(\frac{x+1}{x^2-4x+3}\) is factorized to reveal a common denominator with other fractions in the problem.
Here's a tip for simplification: Look for a common factor in the numerator and denominator. In the provided solution, the numerator didn't have a common factor that could simplify the fraction further, but this step can often lead to a more reduced form of the expression.

Common Denominator

Dealing with multiple fractions almost inevitably leads to finding a common denominator. This is an essential technique in the process of simplifying complex algebraic fractions like the one in the given exercise. A common denominator is a shared multiple of all the denominators involved in the problem. It allows you to combine fractions by ensuring that each term is expressed with the same bottom number (denominator).
In our exercise, the common denominator for the fractions \(\frac{3x+1}{x-1}\), \(\frac{x-1}{x-3}\), and \(\frac{x+1}{x^2-4x+3}\) after factorizing the last term is \(\left(x-1\right)\left(x-3\right)\). To find a common denominator, you can identify the unique factors present in each of the denominators, then multiply them together, taking care to include each factor the maximum number of times it occurs in any of the denominators. This strategy allows combining the fractions leading to a simplified solution.

Combining Like Terms

Once you have a common denominator and have adjusted the numerators accordingly, you will often need to simplify the resulting expression. This is where combining like terms plays a role. Like terms are terms that have identical variable parts—the letters and exponents match—even if the coefficients (the numbers in front of the variables) are different.
The process involves adding or subtracting coefficients of like terms to consolidate the expression. For example, in the solution provided, the numerator contains several terms with the variable \(x\), which are \(3x\), \(8x\), \(2x\), and \(x\). By combining them through addition or subtraction (keeping in mind their signs), you significantly reduce the complexity of the expression. In the simplified form we reach \(2x^2+2x-3\), which is much more manageable. Always remember to look for like terms first in the numerator and then in the denominator to simplify your algebraic fractions.