Problem 66 - Algebra for College Students

Question

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ -2-\frac{5}{4 x-3} $$

Step-by-Step Solution

Answer

The simplest form is \(\frac{-8x + 1}{4x-3}\).

Step 1: Identify the Rational Parts

The expression we are dealing with is \[-2 - \frac{5}{4x-3}\]Here, \(-2\) is a whole number, and \(-\frac{5}{4x-3}\) is a rational expression.

Step 2: Find a Common Denominator

To combine \(-2\) with the rational expression, we need a common denominator. The denominator for \(-\frac{5}{4x-3}\) is \(4x-3\). We convert \(-2\) to a fraction using this denominator:\[-2 = -2 \times \frac{4x-3}{4x-3} = -\frac{2(4x-3)}{4x-3} = -\frac{8x-6}{4x-3}\]

Step 3: Combine the Expressions

With the common denominator established, we can now combine the fractions:\[-\frac{8x-6}{4x-3} - \frac{5}{4x-3} = \frac{-8x + 6 - 5}{4x-3}\]

Step 4: Simplify the Numerator

Simplify the expression in the numerator:\[-8x + 6 - 5 = -8x + 1\]Thus, the combined expression is:\[\frac{-8x + 1}{4x-3}\]

Step 5: Express the Result in Simplest Form

The rational expression \(\frac{-8x + 1}{4x-3}\) cannot be simplified further since there are no common factors between the numerator and the denominator other than 1.

Key Concepts

Common Denominator

When dealing with rational expressions, a common denominator is necessary to add or subtract fractions. Simply put, a common denominator is the same number shared by the denominators of two or more fractions. When we lack a common denominator, it becomes challenging to combine fractions.

In our exercise, we have \( -2 \) as a whole number and \(-\frac{5}{4x-3}\) as a rational expression. To combine them, we need to transform \( -2 \) into a fraction that shares the same denominator as the rational expression \(\frac{5}{4x-3}\).

• Thus, the denominator of the fraction \( -\frac{5}{4x-3} \) is \(4x-3\).
• We convert \( -2 \) into a fraction with this denominator: \(-2 = -2 \times \frac{4x-3}{4x-3} = -\frac{8x-6}{4x-3}\).

By having the same denominator, we have facilitated the process to subtract the fractions.

Simplifying Expressions

Simplifying rational expressions means to reduce the expression into its simplest form. The simplest form of a rational expression is one in which no common factors, other than 1, exist between the numerator and the denominator. Simplifying makes the expression easier to work with and understand.

In the provided exercise, after we find a common denominator and align the terms under it, we obtain the expression:

• \(\frac{-8x + 6 - 5}{4x-3} = \frac{-8x + 1}{4x-3} \)

To simplify this, our main goal is to reduce the numerator by performing arithmetic operations, as shown above. After ensuring no common factors exist in the numerator and denominator, the expression \(\frac{-8x + 1}{4x-3}\) is already in its simplest form.

• Check for any common factors between the numerator and denominator.
• In this case, no further simplification is possible.

Thus, we confirm that the expression cannot be simplified further.

Subtracting Fractions

To subtract fractions, especially when they have the same denominator, you simply subtract the numerators and keep the common denominator unchanged. This process is much easier when you've already established a common denominator for all the fractions involved.

From our original exercise, after obtaining the fractions \(-\frac{8x-6}{4x-3} \) and \(-\frac{5}{4x-3}\), both fractions share a common denominator of \(4x-3\). Subtraction becomes straightforward, as follows:

• Subtract \(6 - 5\) from the numerators \(-8x+6\) and \-5\: \(-8x + 6 - 5 = -8x + 1\).
• Combine the result into a single fraction: \(\frac{-8x + 1}{4x-3} \).

The result \(\frac{-8x + 1}{4x-3}\) is our combined expression. It's important to ensure accuracy when aligning and handling the numerators for correct subtraction.