Problem 23 - Elementary and Intermediate Algebra

Question

Add and simplify the result, if possible. \(\frac{3 x-5}{x-2}+\frac{6 x-13}{x-2}\)

Step-by-Step Solution

Answer

The expression simplifies to 9.

Step 1: Identify the Common Denominator

Notice that both fractions have a common denominator, which is \(x-2\). This allows us to add the fractions directly by combining their numerators.

Step 2: Add the Numerators

Add the numerators of the fractions: \[(3x - 5) + (6x - 13) = 3x - 5 + 6x - 13.\]

Step 3: Simplify the Numerator

Combine like terms in the expression obtained from adding the numerators:\[3x + 6x = 9x,\] and \[-5 - 13 = -18.\] Thus, the numerator becomes \(9x - 18\).

Step 4: Write the Combined Fraction

Place the simplified numerator over the common denominator:\[\frac{9x - 18}{x-2}.\]

Step 5: Simplify the Fraction by Factoring

Factor the numerator:\[9x - 18 = 9(x - 2).\]Write the equation as:\[\frac{9(x-2)}{x-2}.\]

Step 6: Cancel the Common Factors

Since \(x-2\) appears in both the numerator and denominator, cancel this common factor, leaving:\[9.\]

Step 7: Conclusion

The expression simplifies to \(9\).

Key Concepts

Common Denominator

When adding or subtracting fractions, whether they contain numbers or polynomials, finding a common denominator is crucial. The denominator is the number or expression at the bottom of a fraction. It tells us how many parts make up a whole. In the expression \(\frac{3x-5}{x-2}+\frac{6x-13}{x-2}\), both fractions have \(x-2\) as their denominator. Thus, they already share a common denominator, making it easier to combine them.

• A common denominator means that both fractions split the whole in the same way.
• This allows you to directly add the numerators (the numbers on top) without adjusting the fractions further first.

This step simplifies the process significantly because you don't have to perform any more complex operations beforehand. Just add the numerators: \((3x-5)+(6x-13)\), and you're one step closer to solving the problem!

Simplifying Expressions

To simplify an expression means to combine like terms and reduce the expression to its simplest form. In the given exercise, it involves dealing with the numerators of the fractions: \((3x - 5)\) and \((6x - 13)\). Once these numerators are added together, as shown in the previous section, you will simplify the result.

• Combine like terms: \(3x + 6x = 9x\) and \(-5 - 13 = -18\).
• Thus, the expression becomes \(9x - 18\).

Thanks to combining like terms, the polynomial is much simpler now. This step is critical because it sets up the fraction for easier simplification or further manipulation later on. Remember, simplifying is about making the expression as neat and concise as possible!

Factoring

Factoring is breaking down a composite number or expression into a product of its factors. With polynomial fractions, factoring helps in simplifying an expression by canceling out terms. For the polynomial in our exercise, \(9x - 18\) can be factored by finding the greatest common factor (GCF).

• The GCF of \(9x\) and \(18\) is \(9\), so factor \(9\) out: \(9(x - 2)\).

This factorization is extremely useful because the entire expression was over the common denominator \(x-2\), and factoring allows us to simplify the fraction further.
Since \(x-2\) also appears in the denominator, you can cancel it out: leaving you simply with \(9\). Factoring not only simplifies polynomials but is also essential for solving equations and simplifying mathematical expressions further.