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What Are Algebraic Fractions?
An algebraic fraction is just like a regular fraction, but instead of numbers, it contains algebraic expressions in the numerator, denominator, or both.
Examples:
Manipulate Algebraic Fractions
This means:
Add/Subtract: Use a common denominator.
Multiply: Multiply numerators and denominators directly.
Divide: Flip the second fraction (reciprocal) and multiply (keep first fraction, change the sign, flip the second fraction).
Simplify: Cancel common factors after factoring.
Factorize and Simplify Rational Expressions
This means:
Factor both numerator and denominator.
Cancel any common factors.
Watch out for expressions that can’t be simplified!!!
Example: Simplify
Steps:
Factor numerator: 𝑥(𝑥−2)
Factor denominator: (𝑥−2)(𝑥−3)
Cancel common factor: 𝑥−2
Indices:
Indices (also called powers or exponents) tell you how many times to multiply a number or variable by itself. Example:
a² = a x a
Laws:
Example:
Solve 32ˣ = 2
Rewrite 32 as 2⁵
→ (2⁵)ˣ = 2
→ 2⁵ˣ = 2¹ . Now that we have the same base, we can work with the indices alone.
→ 5x = 1 so x = 1⁄5
Solve 5ˣ⁺¹ = 25ˣ
Rewrite 25 as 5²
→ 5ˣ⁺¹ = 5²ˣ. Same base = work with indices alone.
→ x+ 1 = 2x so x = 1
(3x⁻⁴) × (2x²⁄³) × (x¹⁄²)
Step 1: Multiply the coefficients → 3 × 2 = 6
Step 2: Add the powers of x → x⁻⁴ + x²⁄³ + x¹⁄² To add these, convert all powers to a common denominator:
x⁻⁴ = x⁻²⁴⁄⁶
x²⁄³ = x⁴⁄⁶
x¹⁄² = x³⁄⁶
Now add: → x⁻²⁴⁄⁶ + x⁴⁄⁶ + x³⁄⁶ = x⁻¹⁷⁄⁶
Final Answer: 6x⁻¹⁷⁄⁶
Equations:
Solving linear equations (one unknown):
Goal: Isolate the variable (usually x) on one side.
Example 1: 3x + 4 = 10
→ Subtract 4: 3x = 6
→ Divide by 3: x = 2
Example 2: 5 – 2x = 3(x + 7)
→ Expand: 5 – 2x = 3x + 21
→ Rearrange: –2x – 3x = 21 – 5
→ –5x = 16 → x = –16⁄5
Solving fractional equations:
→ Multiply both sides by (2x + 1)
→ x = 4(2x + 1)
→ x = 8x + 4
→ Rearranged: –7x = 4 so x = –4⁄7
Simultaneous Equations (Linear)
Method 1: Substitution
The equations: 2x - y = 3 and x + y = 7
Rewrite: x + y = 7 → y = 7 – x
Substitute into second equation: 2x – (7 – x) = 3 → Solve for x
Method 2: Elimination
Multiply equations to align coefficients, then add/subtract to eliminate one variable.
The equations: 2x - y = 3 and x + y = 7.
You can add both equations since the y variables will cancel out (opposite signs).
This leaves you with 3x = 10 so x=10/3
Solving Quadratic Equations
Method 1: Factorization
x² + 5x + 6 = 0 → (x + 2)(x + 3) = 0 → x = –2 or –3
Method 2: Quadratic formula
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