Problem 15 - Prealgebra and Introductory Algebra

Question

Simplify. $$\frac{x-7}{2 x+7}-\frac{4 x-3}{2 x+7}$$

Step-by-Step Solution

Answer

The simplified result is \(\frac{-3x + 4}{2x + 7}\).

Step 1: Identifying Common Denominators

Inspect the two fractions and notice that both have the same denominator, \(2x+7\). When the denominators are the same, the fractions can be simplified directly.

Step 2: Subtracting Numerators

Subtract the numerators directly. Do \((x-7) - (4x-3)\). This gives \((-3x + 4)\).

Step 3: Writing the Final Result

After performing the subtraction in the numerator, write the result over the original denominator to find the simplified expression. So, the simplified result is \(\frac{-3x + 4}{2x + 7}\).

Key Concepts

Common Denominators

A common denominator is a shared number or expression found in the denominator of two or more fractions. When fractions have the same denominator, it’s much easier to perform operations like addition or subtraction. In the given exercise, both fractions share the denominator \(2x+7\).
This is advantageous because it allows us to focus on the numerators alone when performing subtraction. Remember:

• A common denominator means you don't have to alter the fractions before adding or subtracting them.
• Simply combine the numerators with the operation sign indicated (subtract in this case).

Recognizing common denominators helps streamline simplifying processes in algebra.

Numerators

The numerators are the top parts of the fractions, representing parts of a whole depicted by the denominator. In subtraction, the process is straightforward once the denominators are common. For the given problem, the numerators are \(x-7\) and \(4x-3\).
When denominators are already equal, focus solely on the numerators for further calculations. You simply subtract them directly, respecting the order and signs:

• First term in the first numerator is \(x\).
• Subtracting the entire second numerator \((4x-3)\) requires distributing the subtraction sign.
• This becomes \((x - 4x) + (-7 - (-3))\).

Understanding numerator operations in fractions is crucial for accurate simplification.

Simplification

Simplification in math refers to reducing an expression or equation to its simplest form. In fractions, it means making them easier to work with, often by breaking them down to their lowest terms. The core aim is to maintain the original value but present it in a more manageable format.
For our exercise, after performing the numerator subtraction, the expression is simplified into \(\frac{-3x+4}{2x+7}\). The steps were:

• Subtract \(4x\) from \(x\), resulting in \(-3x\).
• Then subtract \(-3\) from \(-7\), yielding \(4\).

This transformation helps to clearly display the result, making further calculations easier if needed.

Subtraction in Algebra

Subtraction in algebra involves finding the difference between two algebraic expressions. It can involve variables, constants, or both. Critical steps include distributing the subtraction across the terms, and carefully handling like terms.
In the problem given, the subtraction step was crucial:

• Handled by putting quantities such as \(x - 4x\) and \(-7 + 3\).
• Remember to flip the sign of every term in the second expression (\(4x - 3\)) when subtracting.

This keeps the negative and positive impacts correctly in check. Grasping this is key for accuracy in more complex algebraic contexts.