Problem 43 - Algebra 1

Question

SIMPLIFYING RATIONAL EXPRESSIONS Simplify the expression. $$\frac{5}{4 x}-\frac{7}{3 x}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \( \frac{-13}{12x} \).

Step 1: Identifying the Expression

The given expression is \( \frac{5}{4 x} - \frac{7}{3 x} \). This is a rational expression since it consists of two rational fractions.

Step 2: Identifying the Common Denominator

For combining the two fractions, we need a common denominator. The common denominator would be the product of \(4x\) and \(3x\) which is \(12x^2\).

Step 3: Altering the Fractions to have the Common Denominator

We can express the fractions having \(12x^2\) as their denominator. To do this, we multiply top and bottom of \( \frac{5}{4x} \) by \(3x\) and \( \frac{7}{3x} \) by \(4x\). So, the altered expressions are \( \frac{15x}{12x^2} \) and \( \frac{28x}{12x^2} \) respectively.

Step 4: Substituting the Fractions in the Expression

Now substitute these altered fractions into the given expression. \( \frac{15x}{12x^2} - \frac{28x}{12x^2} = \frac{15x-28x}{12x^2} = \frac{-13x}{12x^2} \).

Step 5: Simplifying the Expression

Simplify the above expression by cancelling the \(x\) factor out from numerator and denominator to get \( \frac{-13}{12x} \). This is the final simplified expression.

Key Concepts

Common Denominator

When simplifying rational expressions like \( \frac{5}{4x} - \frac{7}{3x} \), finding a common denominator is crucial. A common denominator is a common multiple of the original denominators. It allows us to rewrite the fractions so we can easily subtract or add them.

For this expression, we aim to consolidate \( 4x \) and \( 3x \) into a more manageable term.

• Start by finding the least common multiple (LCM) of the coefficients 4 and 3, which is 12.
• For the variable part, \( x \), you simply multiply to form \( x^2 \).

Hence, the common denominator is \( 12x^2 \). By achieving this, we've created a foundation that allows straightforward algebraic manipulation.

Rational Fractions

Rational fractions are fractions where both the numerator and the denominator are polynomials. They can seem a bit intimidating at first glance, but by manipulating them smartly, they become much easier to handle.

Consider our expression, \( \frac{5}{4x} - \frac{7}{3x} \). It consists of two rational fractions. Here, understanding how to work with each component is key.

• The numerator is a simple integer in both fractions: 5 and 7, respectively.
• The denominators consist of both a constant and a variable: \( 4x \) and \( 3x \).

By addressing both parts of these fractions, we can perform operations more fluidly, such as subtraction in this case.

Algebraic Manipulation

Algebraic manipulation is all about reshaping and simplifying expressions to make them easier to process. Once we have a common denominator like \( 12x^2 \), algebraic tactics enable us to rewrite and simplify rational expressions. Here’s how to perform algebraic manipulation for subtraction:

With fractions \( \frac{5}{4x} \) and \( \frac{7}{3x} \) rewritten over the common denominator \( 12x^2 \):

• Multiply \( \frac{5}{4x} \) by \( \frac{3x}{3x} \) to obtain \( \frac{15x}{12x^2} \).
• Then, multiply \( \frac{7}{3x} \) by \( \frac{4x}{4x} \) to get \( \frac{28x}{12x^2} \).

Now you can subtract the numerators directly: \( 15x - 28x = -13x \). So the resulting expression is \( \frac{-13x}{12x^2} \). Simplifying further by cancelling out the \( x \) gives us \( \frac{-13}{12x} \). This manipulation helps in removing complexities and reaching a simplified solution.