Problem 93 - Intermediate Algebra

Question

Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\frac{2 a+1}{3 a-2}-\frac{a-4}{2-3 a}$$

Step-by-Step Solution

Answer

\(\frac{3(a-1)}{3a-2}\) is the simplified expression.

Step 1: Identify the common denominators

Analyze the denominators of the expressions \(3a - 2\) and \(2 - 3a\). Notice that \(2 - 3a\) is the negative of \(3a - 2\) because \(2 - 3a = -(3a - 2)\).

Step 2: Adjust the expressions to a common denominator

To make \(\frac{a-4}{2-3a}\) compatible with \(\frac{2a+1}{3a-2}\), multiply both the numerator and the denominator by -1 to get \(\frac{-(a-4)}{3a-2}\). This gives us \(\frac{-a+4}{3a-2}\).

Step 3: Combine the expressions

Now, both expressions are over a common denominator \(3a-2\), so we can combine them: \(\frac{2a+1}{3a-2} - \frac{-a+4}{3a-2} = \frac{2a+1 + a - 4}{3a-2}\).

Step 4: Simplify the numerator

Simplify the numerator by combining like terms: \(2a + a + 1 - 4 = 3a - 3\). The expression is now \(\frac{3a - 3}{3a-2}\).

Step 5: Check if further simplification is possible

Check if \(3a - 3\) can be factored and if any terms can be canceled with the denominator. Factor the numerator: \(3(a-1)\), so we have \(\frac{3(a-1)}{3a-2}\). No further simplification can be done, as \(3a-2\) has different terms.

Key Concepts

Addition and Subtraction of Fractions

When adding or subtracting fractions, the goal is to combine them into a single fraction. This process involves ensuring both fractions have a common denominator. The numerator is adjusted accordingly without changing the fraction's value.

• Ensure the denominators are the same. If not, find a common denominator.
• Adjust the numerators to reflect changes made to the denominators.
• Combine numerators through addition or subtraction.

Understanding how the numerator and denominator work together makes these operations easier. It's crucial to adjust the numerators correctly so the final value of the fraction remains the same.

Common Denominator

To perform operations on fractions, especially addition and subtraction, having a common denominator is essential. A common denominator allows fractions to be directly compared and combined.

• The denominator provides the "parts" of the whole. For comparison, these parts must match.
• If denominators differ, they must be adjusted to become equal. This may involve multiplication or finding their least common multiple.
• Notice signs or factors in the denominator when performing adjustments, like reversing a sign to match another expression.

Finding a common denominator prevents errors in the computation and helps streamline the process of combining fractions.

Simplification of Algebraic Expressions

Simplification involves making an expression as straightforward as possible. After adding or subtracting, simplifying the resultant fraction often reveals its essence.

• Combine like terms to reduce redundancy.
• Simplify numerators and denominators before assuming an expression is fully simplified.
• Watch for opportunities to factor out common terms.

The simplification process not only makes the expression easier to understand but is crucial for avoiding miscalculations, ensuring that the expression is in its best form.

Factoring

Factoring is breaking down an expression into products of simpler expressions. It simplifies processes like simplification by uncovering potential cancellations.

• Identify terms with common factors in both numerator and denominator.
• Factor them out to see if they provide easier expressions to work with or cancel out.
• Factoring can reveal deep structural insights into the expression's composition.

Understanding factoring makes dealing with complex expressions more manageable, as it reduces opacity and offers simplicity.