Simplifying Rational Expressions by Adding or Subtracting with Like Denominators

Overview

• When adding or subtracting rational expressions with like denominators, they can be combined into a single fraction.
• The process involves combining numerators while keeping the common denominator.

Key Steps in Simplification

1. Combine Numerators: Add or subtract the numerators as indicated.
2. Factor: Simplify the resulting expression by factoring out any common factors.
3. Reduce: Cancel out any common terms between numerator and denominator if possible.

Examples

Example 1

• Expression: ( \frac{3x}{5} + \frac{4x + 7}{5} )
• Step 1: Combine into a single fraction: ( \frac{3x + 4x + 7}{5} )
• Step 2: Simplify the numerator:
  • ( 3x + 4x = 7x )
• Step 3: Optionally factor out the GCF in the numerator:
  • ( 7(x + 1) )
• Final Answer: ( \frac{7(x + 1)}{5} )

Example 2

• Expression: ( \frac{7}{x} - \frac{11}{x} )
• Step 1: Combine into a single fraction: ( \frac{7 - 11}{x} )
• Step 2: Simplify the numerator:
  • ( 7 - 11 = -4 )
• Final Answer: ( \frac{-4}{x} )

Example 3

• Expression: ( \frac{7x + 4}{x + 2} + \frac{5x - 7}{x + 2} )
• Step 1: Combine into a single fraction:
  • ( \frac{7x + 5x + 4 - 7}{x + 2} )
• Step 2: Simplify the numerator:
  • ( 7x + 5x = 12x )
  • ( 4 - 7 = -3 )
• Step 3: Optionally factor:
  • ( 3(4x - 1) )
• Final Answer: ( \frac{4x - 1}{x + 2} )

Example 4

• Expression: ( \frac{3x - 10}{x - 1} - \frac{5x - 12}{x - 1} )
• Step 1: Write as a single fraction:
  • ( \frac{3x - 10 - (5x - 12)}{x - 1} )
• Step 2: Distribute the negative sign:
  • ( \frac{3x - 10 - 5x + 12}{x - 1} )
• Step 3: Simplify the numerator:
  • ( 3x - 5x = -2x )
  • ( -10 + 12 = 2 )
• Step 4: Factor GCF in the numerator:
  • ( -2(x - 1) )
• Final Answer: Cancel ( x - 1 ):
  • (-2)

Conclusion

• Combining rational expressions with like denominators is straightforward when applying arithmetic operations to the numerators and keeping the common denominator.
• Always check for opportunities to factor and reduce the expression to its simplest form.