Problem 65 - Algebra for College Students

Question

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ -1-\frac{3}{2 x+1} $$

Step-by-Step Solution

Answer

The simplest form is \(\frac{-2x - 4}{2x + 1}\).

Step 1: Identify the Terms

First, note that we are asked to subtract a rational expression from a whole number. Our expression is \[-1 - \frac{3}{2x+1}\].

Step 2: Rewrite Whole Number as Fraction

Convert the whole number \(-1\) into a fraction with the same denominator as the other term. Since the denominator of \(\frac{3}{2x+1}\) is \(2x+1\), rewrite \(-1\) as \(-\frac{2x+1}{2x+1}\). This simplifies the subtraction process.

Step 3: Perform the Subtraction

Now subtract the two fractions with a common denominator: \[-\frac{2x+1}{2x+1} - \frac{3}{2x+1} = \frac{-(2x+1) - 3}{2x+1}\].

Step 4: Simplify the Numerator

Expand and simplify the numerator: \[-(2x+1) - 3 = -2x - 1 - 3 = -2x - 4\].

Step 5: Write the Simplified Expression

The expression simplifies to: \[\frac{-2x - 4}{2x+1}\].

Step 6: Factor the Numerator and Simplify Further if Possible

Attempt to factor the numerator \(-2x - 4\) to see if it can be further simplified. It can be factored as \(-2(x + 2)\). However, since there is no common factor in the numerator and denominator \(2x+1\), this is the simplest form.

Key Concepts

Simplifying Expressions

Simplifying expressions is a foundational skill in algebra that revolves around making algebraic expressions as straightforward as possible, while still meaning the same thing. Imagine you have a long, confusing thought. Simplifying it is like finding a concise way to express that thought clearly. It involves reducing expressions without altering their value.

To simplify:

• Combine like terms - these are terms with the same variable raised to the same power.
• Use distributive property - a way to multiply a number by a sum or difference.
• Reduce fractions to their simplest form - divide both numerator and denominator by their greatest common factor.

In our example, the expression \[-1 - \frac{3}{2x+1}\] was simplified by rewriting the whole number in terms of the common denominator, helping the subtraction process. Simplifying the expression means not only finding a result but expressing it in the clearest way possible.

Subtracting Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. Subtracting these expressions follows similar steps to subtracting regular fractions.

Here's how you can do it:

• Find a common denominator for the fractions involved.
• Rewrite each fraction with the common denominator.
• Subtract the numerators while keeping the common denominator the same.
• Simplify the result, if possible, by factoring and reducing.

In our exercise, the subtraction was carried out by first rewriting \(-1\) as a fraction with the denominator \(2x+1\), allowing the terms to be combined into a single expression. The subtraction itself is often straightforward once the fractions share a common denominator.

Algebraic Fractions

Much like regular fractions, algebraic fractions involve division, with a key difference. They include variables in the numerator, denominator, or both. Understanding them is vital in algebra as they lay the groundwork for more complex operations.

Key points about algebraic fractions:

• The numerator and denominator are polynomials.
• You can only subtract them if they have a common denominator, similar to numerical fractions.
• Complex algebraic fractions might need simplification, involving factoring polynomials.
• Always check if a fraction can be reduced by dividing both parts by their greatest common factor.

In our equation, separating and dealing with the fraction element involved understanding that \(\frac{3}{2x+1}\) cannot stand alone like a whole number—it needs pieces common to both terms for subtraction. Simplifying these fractions often involves reducing or factoring.