Problem 11 - Algebra for College Students

Question

Perform the indicated operations. Reduce answers to their lowest terms. See Example \(I\) $$ \frac{x-3}{2 x}-\frac{3 x-5}{2 x} $$

Step-by-Step Solution

Answer

\(\frac{-x + 1}{x}\)

Step 1: Identify Common Denominator

Observe that both fractions have a common denominator of \(2x\). This means we can combine the numerators directly.

Step 2: Combine the Numerators

Since the denominators are the same, subtract the numerators: \[\frac{(x-3) - (3x-5)}{2x} = \frac{x - 3 - 3x + 5}{2x}\]

Step 3: Simplify the Numerator

Combine like terms in the numerator: \[x - 3 - 3x + 5 = (x - 3x) + (-3 + 5) = -2x + 2\]

Step 4: Express Simplified Fraction

After simplifying the numerator, rewrite the fraction: \[\frac{-2x + 2}{2x}\]

Step 5: Reduce to Lowest Terms

Factor out the common factor in the numerator: \[\frac{2(-x + 1)}{2x}\] Cancel the common factor of 2: \[\frac{-x+1}{x}\] This can also be written as: \[1 - \frac{x}{x} = 1 - 1 = 0\]

Key Concepts

fraction subtraction

When dealing with fraction subtraction, it's essential to ensure that the fractions have a common denominator. Subtracting fractions involves working with their numerators while keeping the denominator the same.
The given problem involves the fractions: \( \frac{x-3}{2x} \) and \( \frac{3x-5}{2x} \). Since both fractions share the common denominator \( 2x \), we can directly subtract the numerators:
\[ \frac{(x-3)-(3x-5)}{2x} \]
Lift the parentheses in the numerator carefully while keeping the denominator unchanged:
\[ \frac{x - 3 - 3x + 5}{2x} \]
This ensures we have accurately captured the required operation.

common denominators

Finding a common denominator is crucial for adding and subtracting fractions.
In this problem, the denominators are already the same (both \(2x\)).
With a shared common denominator, you do not need to adjust the denominations of the fractions further.
When the denominators are different, find the least common multiple (LCM) of the denominators to continue with the arithmetic operations needed.
However, since both fractions already share \(2x\) as a common denominator in this question, it's straightforward:
\[ \frac{x - 3}{2x} - \frac{3x - 5}{2x} \]
becomes:
\[ \frac{x - 3 - (3x - 5)}{2x} \]
This simplifies the numerator which then can be reduced further.

simplifying algebraic expressions

Simplifying algebraic expressions involves combining like terms and reducing fractions if possible.
The equation we have is:
\[ \frac{x - 3 - 3x + 5}{2x} \]
First, combine like terms:
\[ (x - 3x) + (-3 + 5) = -2x + 2 \]
This changes our fraction to:
\[ \frac{-2x + 2}{2x} \]
Next, factor out common terms in the numerator:
\[ \frac{2(-x + 1)}{2x} \]
The 2's cancel each other out:
\[ \frac{-x + 1}{x} \]
This can further simplify to:
\[ 1 - \frac{x}{x} = 1 - 1 = 0 \]
Therefore, mastering simplifying algebraic expressions lets you check if your resultant terms can reduce to their lowest terms. It’s crucial for getting the correct answer.