Problem 39 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The simplified form of the given expression is \(-x^2 - 14x + 2\).

Step 1: Find the Least Common Denominator

The denominators here are \(3x - 1\) and \(x - 2\). The least common denominator (LCD) is the product of these two denominators, which is \( (3x - 1)(x - 2)\).

Step 2: Rewrite the fractions with the LCD

Rewrite the expression using the LCD as the denominator for each fraction: \[\frac{(2x + 1)(x - 2)}{(3x - 1)(x - 2)} - \frac{(x + 4)(3x - 1)}{(3x - 1)(x - 2)}\]

Step 3: Simplify the Numerator

Now let's simplify the numerators in these two terms:The first term's numerator simplifies to \(2x^2 - 4x + x - 2 = 2x^2 - 3x - 2\).The second term's numerator simplifies to \(3x^2 - x + 12x - 4 = 3x^2 + 11x - 4\).So the expression becomes:\[\frac{2x^2 - 3x - 2}{(3x - 1)(x - 2)} - \frac{3x^2 + 11x - 4}{(3x - 1)(x - 2)}\]

Step 4: Subtract the two fractions

Since both fractions have the same denominator, they can be subtracted. This means subtracting each term in the numerator of the second fraction from the corresponding term in the numerator of the first fraction. \[\frac{(2x^2 - 3x - 2) - (3x^2 + 11x - 4)}{(3x - 1)(x - 2)} = \frac{-x^2 - 14x + 2}{(3x - 1)(x - 2)}\]

Key Concepts

Least Common Denominator

When working with rational expressions, finding the Least Common Denominator (LCD) is often essential. The LCD is the smallest expression divisible by all denominators in the problem. To find the LCD for this problem, we examine the denominators: \(3x - 1\) and \(x - 2\). The LCD is simply the product of these denominators: \((3x - 1)(x - 2)\).
This method ensures both fractions have a common base, making them easier to subtract or add.

• The purpose of finding an LCD is to eliminate complex fraction conversions later.
• Multiply each fraction's numerator by the complementary part of the LCD that does not originally include it.

Remember, while the LCD might seem larger, it's necessary to align the fractions for straightforward operations like subtraction.

Numerator Simplification

Numerator simplification involves breaking down and factoring expressions in the numerator to make the subtraction process manageable. Start with expanding each numerator using distributive or FOIL methods and then combine like terms.
For the expression \(\frac{(2x + 1)(x - 2)}{(3x - 1)(x - 2)}\), it simplifies to \(2x^2 - 3x - 2\).

• Combine like terms: It's essential to add or subtract coefficients of similar variables.
• Organize terms: Typically, arrange from the highest degree to the lowest.

For instance, \(2x^2\) from \(2x^2 - 4x + x - 2\) works by combining the \(-4x + x\) to produce \(-3x\). Repeat similar steps for any rational expression.
Effective simplification helps minimize errors during subtracting or further manipulation of algebraic expressions.

Subtracting Rational Expressions

Subtracting rational expressions is straightforward if both expressions have the same denominator. In this exercise, our common denominator \((3x - 1)(x - 2)\) lets us directly subtract the numerators.
This part of the process involves carefully subtracting each corresponding term: \((2x^2 - 3x - 2) - (3x^2 + 11x - 4)\).

• Perform operations between like terms: Subtraction is akin to adding the opposite.
• Keep track of negative signs: They can change the result entirely if mishandled.

After combining terms, the new numerator becomes \(-x^2 - 14x + 2\).
Ensuring you're meticulous in calculating even the simplest operations can prevent mistakes and ensures your final expression is accurate.