Chapter 1
Linear Equations and Graphing
Course: Gre
Subject: Gre Quantitative Reasoning
Chapter: Rational Expressions And Equations
Introduction: Applying the Steps
Adding and subtracting rational expressions often requires factoring, finding common denominators, and simplifying. This lesson provides additional practice with more complex examples.
Example 1: Factoring First
Simplify: x+43x-9+x-56x-18
Step 1: Factor denominators
3x-9=3(x-3) 6x-18=6(x-3)
Step 2: LCD = 6(x-3) Step 3: Rewrite first fraction with denominator 6(x-3):
x+43(x-3)×22=2(x+4)6(x-3)
Step 4: Combine:
2(x+4)+(x-5)6(x-3)=2x+8+x-56(x-3)=3x+36(x-3)
Step 5: Factor numerator: 3(x+1)
3(x+1)6(x-3)=x+12(x-3)
Result: x+12x-6(since 2(x-3)=2x-6)
Example 2: Factoring Trinomials
Simplify: x-2x+5+x2+5x+6x2+8x+15
Step 1: Factor
x2+5x+6=(x+3)(x+2) x2+8x+15=(x+5)(x+3)
Step 2: LCD = x+5)(x+3 Step 3: Rewrite:
x-2x+5×x+3x+3=x-2)(x+3x+5)(x+3 Second fraction: x+3)(x+2x+5)(x+3
Step 4: Combine:
x-2)(x+3)+(x+3)(x+2x+5)(x+3
Step 5: Factor x3from numerator:
(x+3)[(x-2)+(x+2)]x+5)(x+3=x+3)(2xx+5)(x+3
Step 6: Cancel x3:
2xx+5
Key Takeaways
- Always factor denominators first
- Find the LCD using the highest powers of each factor
- Rewrite each fraction with the LCD
- Combine numerators and simplify
- Factor and cancel when possible