10TH GRADE · EM2 · RATIONALS

Rational Expressions & Equations

Simplify, Operations & Solving

14 slides LCD Test Prep

What Is a Rational Expression?

P(x) / Q(x)

Examples

Denominator ≠ 0!

x ≠ 1

Simplifying

1. Factor both

2. Cancel common

3. Excluded values

(x²−4)/(x+2)

Factor

x−2, x≠−2

Multiply & Divide

Multiply: AC/BD

Divide: flip & multiply

Factor → Cancel → Multiply

Example

x−3

LCD

LCD = Least Common Denominator

1. Factor denoms

2. Highest power of each

3. Multiply

Denoms: x(x+1), (x+1)(x−2)

LCD = x(x+1)(x−2)

Add & Subtract

Same denom: add tops

Diff denoms: LCD first

3/x + 5/(x+2)

LCD = x(x+2)

Rewrite

(8x+6)/[x(x+2)]

Subtraction

2x/(x−3) − 4/(x+1)

LCD

Rewrite

Expand

Simplify

Distribute the negative!

Solving Rational Equations

1. Find LCD

2. Multiply by LCD

3. Solve

4. CHECK!

Solving Example

3/(x−2) + 1/x = 5/[x(x−2)]

LCD = x(x−2)

3x + x−2 = 5

x = 7/4

Valid!

Extraneous Solutions

x/(x−3) = 3/(x−3) + 2

Multiply by LCD

x = 3

x=3 is EXTRANEOUS

Always check!

Complex Fractions

Fraction within fraction

Multiply by inner LCD

Example

LCD = xy

(x+y)/(y−x)

Practice

1. Simplify

▶ Show Answer

(x+2)/(x+3)

2. Add

▶ Show Answer

(5x+1)/[(x+1)(x−1)]

3. Solve

▶ Show Answer

2±√6

4. Divide

▶ Show Answer

x(x+1)

5. Check extraneous!

▶ Show Answer

x=0

Common Mistakes

Cancel FACTORS not terms

Always check answers!

Distribute the negative

State excluded values

Flip the second!

Rationals Summary

✔ Simplify

✔ Multiply

✔ Divide

✔ Add/Subtract

✔ Solve & check

✔ Domain restrictions

Good luck!