Digital SAT Maths -Unit 2 - 2.3 Rational Expressions- Study Notes- New Syllabus
Digital SAT Maths -Unit 2 – 2.3 Rational Expressions- Study Notes- New syllabus
Digital SAT Maths -Unit 2 – 2.3 Rational Expressions- Study Notes – per latest Syllabus.
Key Concepts:
Simplifying algebraic fractions
Restrictions (denominator ≠ 0)
Simplifying Algebraic Fractions
A rational expression is a fraction that contains a variable in the numerator, denominator, or both.
\( \dfrac{x+2}{x-3} \), \( \dfrac{3x^2}{x+5} \)
To simplify a rational expression, we cancel common factors. This is exactly the same idea as simplifying numerical fractions.
Key Rule
You may only cancel factors, not terms.
Correct: \( \dfrac{2x}{2}=x \)
Incorrect: \( \dfrac{x+2}{x} = 2 \) (cannot cancel across addition)
Steps to Simplify
- Factor numerator
- Factor denominator
- Cancel common factors
DIGITAL SAT Tip
Many SAT problems are testing factoring, not fractions. If a rational expression looks complicated, factor first.
Example 1:
Simplify \( \dfrac{6x}{3} \).
▶️ Answer/Explanation
\( \dfrac{6x}{3}=2x \)
Answer: \( 2x \)
Example 2:
Simplify \( \dfrac{x^2-9}{x-3} \).
▶️ Answer/Explanation
Factor numerator:
\( x^2-9=(x-3)(x+3) \)
\( \dfrac{(x-3)(x+3)}{x-3} \)
\( =x+3 \)
Answer: \( x+3 \)
Example 3:
Simplify \( \dfrac{2x^2+4x}{2x} \) for \( x\ne0 \).
▶️ Answer/Explanation
Factor numerator:
\( 2x(x+2) \)
\( \dfrac{2x(x+2)}{2x} \)
\( =x+2 \)
Answer: \( x+2 \)
Restrictions (Denominator ≠ 0)
A rational expression is undefined when the denominator equals 0.
Therefore, certain values of the variable are not allowed. These are called restrictions on the domain.
Main Rule
Denominator \( \ne 0 \)
To find restrictions:
- Set the denominator equal to 0
- Solve for the variable
- Exclude those values
Important SAT Idea
Even if a factor cancels when simplifying, the restriction still remains.
This is one of the most common DIGITAL SAT trap questions.
Example 1:
Find the restriction for \( \dfrac{1}{x-5} \).
▶️ Answer/Explanation
\( x-5=0 \)
\( x=5 \)
Restriction: \( x\ne5 \)
Example 2:
Find the restrictions for \( \dfrac{2}{x^2-9} \).
▶️ Answer/Explanation
Factor denominator:
\( x^2-9=(x-3)(x+3) \)
\( x=3 \) or \( x=-3 \)
Restrictions: \( x\ne3,\; x\ne-3 \)
Example 3:
Find the restriction for \( \dfrac{x-2}{x-2} \).
▶️ Answer/Explanation
Even though the expression simplifies to 1, the denominator still cannot be 0.
\( x-2=0 \Rightarrow x=2 \)
Restriction: \( x\ne2 \)