Operations on Rational Expressions Simplified

 Operations on Rational Expressions

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Introduction 

Hook: Imagine you're planning a road trip and need to calculate the average speed for different segments of your journey. If you travel $\frac{100}{x+2}$ miles in the first hour and $\frac{150}{x-3}$ miles in the second hour, how would you find the total distance per hour? This requires operations on rational expressions—let's learn how!

Objective:
By the end of this lesson, you'll be able to:

• Add, subtract, multiply, and divide rational expressions.

• Simplify complex rational expressions.

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Prerequisite Knowledge Check

Before we start, ensure you're familiar with:

1. Simplifying fractions (e.g., $\frac{6}{8}=\frac{3}{4}$).

2. Factoring polynomials [e.g., $x^2-5x+6=(x-2)(x-3)$].

3. Finding the Least Common Denominator (LCD) (e.g., LCD of $3$ and $4$ is 12).

Quick Review: Math is Fun – Factoring Quadratics

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Core Concept Explanation (I Do – Teacher alone modeling)

Definition: A rational expression is a fraction where the numerator and denominator are polynomials (e.g., $\frac{3x}{x^2-4}$).

Adding Rational Expressions

Problem: Add $\frac{2}{x+1}+\frac{3}{x-2}$.

Step 1: Find the LCD

• Denominators: $(x+1)$ and $(x-2)$.

• LCD = $(x+1)(x-2)$.

Step 2: Rewrite Each Fraction

• $\frac{2}{x+1}$ becomes $\frac{2(x-2)}{(x+1)(x-2)}$.

• $\frac{3}{x-2}$ becomes $\frac{3(x+1)}{(x+1)(x-2)}$.

Step 3: Add the Numerators

$\frac{2(x-2)+3(x+1)}{(x+1)(x-2)}$= $\frac{2x-4+3x+3}{(x+1)(x-2)}$= $\frac{5x-1}{(x+1)(x-2)}$.

Final Answer: $\frac{5x-1}{(x+1)(x-2)}$

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Guided Practice (We Do – Teacher & Students Together)

Problem: Subtract $\frac{4}{y+3}-\frac{1}{y-1}$.

Prompts:

1. What's the LCD of $(y+3)$ and $(y-1)$?

   • (Answer: $(y+3)(y-1)$
     

2. How do we rewrite the first fraction?

   • (Answer: $\frac{4(y-1)}{(y+3)(y-1)}$
     

3. What's the final simplified form?

   • (Answer: $\frac{4(y-1)-1(y+3)}{(y+3)(y-1)}=\frac{3y-7}{(y+3)(y-1)}$
     

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Independent Practice (You Do – Students Try Alone)

Pause and solve these before checking the solutions!

1. Multiply: $\frac{2x}{x+4}\times \frac{x^2-16}{5x}$

   • (Hint: Factor $x^2-16$ first!)

   • Solution: $\frac{2x(x-4)(x+4)}{5x(x+4)}=\frac{2(x-4)}{5}$

2. Divide: $\frac{3a}{a-2}÷\frac{a+2}{a^2-4}$

   • (Hint: Flip and multiply!)

   • Solution: $\frac{3a(a-2)(a+2)}{(a-2)(a+2)}=3a$
     

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Common Mistakes & Troubleshooting

• Mistake 1: Forgetting to factor first [e.g., $x^2-9$ is $(x+3)(x-3)$].

• Mistake 2: Cancelling terms (e.g., $\frac{x+2}{x+3}$ can't be simplified further!).

• Tip: Always check for excluded values (denominator ≠ 0).

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Few Real-World Applications

Engineering Example: Rational expressions model resistance in parallel circuits:
$\frac{1}{R_{total}}=\frac{1}{R_1}+\frac{1}{R_2}$. Mastering operations helps design efficient systems!

Kinematics (Average Speed)

If a car travels 50 km at 60 km/h and another 50 km at 40 km/h:

$$\text{Average Speed}=\frac{50+50}{\frac{50}{60}+\frac{50}{40}}=\frac{100}{\frac{5}{6}+\frac{5}{4}}=48\text{ km/h}$$

Economics

a. Average Cost Function

Average cost per unit:

$$\text{Average Cost}=\frac{C(x)}{x}=\frac{500+10x}{x}=\frac{500}{x}+10$$

Medicine

a. Drug Concentration in Blood

Concentration $C(t)$ over time:

$$C(t)=\frac{5t}{t^2+1}$$

b. Medical Dosage (Young’s Rule)

Child’s dose:

$$\text{Child’s Dose}=\frac{A}{A+12}\times \text{Adult Dose}$$

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Important things to note

• LCD is key for adding/subtracting.

• Factor first to simplify multiplication/division.

• Always state excluded values (e.g., $x\mathrm{\ne }-1$ in $\frac{1}{x+1}$).

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Practice & Extension

Extra Problems:

1. Add: $\frac{5}{2x}+\frac{3}{x^2}$

   • Solution: $\frac{5x+6}{2x^2}$

Challenge Question:

Simplify: $\frac{\frac{1}{x+h}-\frac{1}{x}}{h}$

• Solution: $\frac{-1}{x(x+h)}$

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Further Resources

• Video Tutorial: Operations on Rational Expressions

• Interactive Practice: Desmos Rational Expressions

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Follow-up discussion 

Subtracting with Unlike Denominators (Advanced Factoring)

Problem: Subtract $\frac{3x}{x^2-9}-\frac{2}{x^2+4x+3}$

Teacher's Step-by-Step:

1. Factor Denominators:

   • $x^2-9=(x+3)(x-3)$
     

   • $x^2+4x+3=(x+1)(x+3)$
     

2. Identify LCD:

   • LCD = $(x+3)(x-3)(x+1)$

3. Rewrite Fractions:

   • First term: $\frac{3x(x+1)}{(x+3)(x-3)(x+1)}$

   • Second term: $\frac{2(x-3)}{(x+3)(x-3)(x+1)}$

4. Subtract & Simplify:

   $$\frac{3x(x+1)-2(x-3)}{(x+3)(x-3)(x+1)}=\frac{3x^2+3x-2x+6}{(x+3)(x-3)(x+1)}=\frac{3x^2+x+6}{(x+3)(x-3)(x+1)}$$

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Multiplying with Cancellation (Variables in Both Terms)

Problem: Multiply $\frac{x^2-4}{x^2+3x+2}\times \frac{x+1}{x-2}$

Teacher's Step-by-Step:

1. Factor All Expressions:

   • $x^2-4=(x+2)(x-2)$
     

   • $x^2+3x+2=(x+1)(x+2)$
     

2. Rewrite Multiplication:

   $$\frac{(x+2)(x-2)}{(x+1)(x+2)}\times \frac{x+1}{x-\mathrm{2}}$$

3. Cancel Common Factors:

   • $(x+2)$ and $(x+1)$ cancel out.

   • $(x-2)$ cancels with the denominator.

4. Final Answer: $1$ (All terms cancelled out)

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Dividing with Complex Fractions

Problem: Divide $\frac{\frac{x}{x-1}}{\frac{x+2}{x^2-1}}$

Teacher's Step-by-Step:

1. Rewrite as Multiplication:

   $$\frac{x}{x-1}\times \frac{x^2-1}{x+\mathrm{2}}$$

2. Factor Difference of Squares:

   • $x^2-1=(x+1)(x-1)$
     

3. Multiply & Simplify:

   $$\frac{x(x+1)(x-1)}{(x-1)(x+2)}=\frac{x(x+1)}{x+\mathrm{2}}$$

Common Mistake Alert: Emphasize that $\frac{x-1}{x-1}=1$only when $x\mathrm{\ne }1$.

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Adding with Binomial Numerators

Problem: Add $\frac{x+1}{x^2-5x+6}+\frac{2x-3}{x^2-4}$

Teacher's Step-by-Step:

1. Factor Denominators:

   • $x^2-5x+6=(x-2)(x-3)$
     

   • $x^2-4=(x+2)(x-2)$
     

2. Find LCD:

   • LCD = $(x-2)(x-3)(x+2)$
     

3. Adjust Numerators:

   • First term: $\frac{(x+1)(x+2)}{(x-2)(x-3)(x+2)}$

   • Second term: $\frac{(2x-3)(x-3)}{(x-2)(x-3)(x+2)}$

4. Combine & Expand:

   $$\frac{x^2+3x+2+2x^2-9x+9}{(x-2)(x-3)(x+2)}=\frac{3x^2-6x+11}{(x-2)(x-3)(x+2)}$$

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Simplifying Complex Rational Expressions

Problem: Simplify $\frac{\frac{1}{x+h}-\frac{1}{x}}{h}$

Teacher's Step-by-Step:

1. Combine Numerator Fractions:

   $$\frac{\frac{x-(x+h)}{x(x+h)}}{h}=\frac{\frac{-h}{x(x+h)}}{\mathrm{h}}$$

2. Divide by $h$:

   $$\frac{-h}{x(x+h)}\times \frac{1}{h}=\frac{-1}{x(x+h)}$$

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Practice problems with Partial solutions. Click those blanks or question marks to write your answer.

1. Missing Numerator (Multiplication)

Problem:

$\frac{3}{x+2}\times \frac{\boxed{{\displaystyle <a\; href="https://www.blogger.com/comment/fullpage/post/9093914164408134262/6198245625841471370"\; target="\_blank">?</a>}}}{x-1}=\frac{6}{(x+2)(x-1)}$

Clues:

• The denominators match on both sides.

• What number × 3 = 6?

• Missing Answer: $\boxed{{\displaystyle \mathrm{<span\; style="color:\; \#cccccc;">2</span>}}}$
  

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Missing Denominator (Addition)

Problem:

$\frac{2}{x}+\frac{3}{\boxed{{\displaystyle <a\; href="https://www.blogger.com/comment/fullpage/post/9093914164408134262/6198245625841471370"\; target="\_blank">?</a>}}}=\frac{2x+3}{x}$

Clues:

• The LCD is just $x$.

• The second denominator must be ______.

• Missing Answer: $\boxed{{\displaystyle \mathrm{<span\; style="color:\; \#cccccc;"><a\; href="https://www.blogger.com/comment/fullpage/post/9093914164408134262/6198245625841471370"\; target="\_blank">x</a></span>}}}$
  

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Missing Factor (Simplification)

Problem:

$\frac{x^2-9}{x+3}=\frac{(x+3)(\boxed{{\displaystyle ?}})}{x+3}=\boxed{{\displaystyle <a\; href="https://www.blogger.com/comment/fullpage/post/9093914164408134262/6198245625841471370"\; target="\_blank">?</a>}}$

Clues:

• Factor $x^2-\mathrm{9\; first.}$

• Cancel the common term.

• Missing Answers:

1. $\boxed{{\displaystyle x-3}}$ and 

2. $\boxed{{\displaystyle \mathrm{<span\; style="color:\; \#cccccc;">x</span>}<span\; style="color:\; \#cccccc;">-</span>\mathrm{<span\; style="color:\; \#cccccc;">3</span>}}}$
   

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Missing Term (Subtraction)

Problem:

$\frac{5}{x}-\frac{\boxed{{\displaystyle <a\; href="https://www.blogger.com/comment/fullpage/post/9093914164408134262/6198245625841471370"\; target="\_blank">?</a>}}}{x}=\frac{5-2}{x}$

Clues:

• The denominators are the same.

• What number subtracted from 5 gives 3?

• Missing Answer: $\boxed{{\displaystyle \mathrm{<span\; style="color:\; \#cccccc;">2</span>}}}$
  

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Missing Divisor (Division)

Problem:
$\frac{4}{x-1}÷\frac{\boxed{{\displaystyle ?}}}{x+2}=\frac{4(x+2)}{(x-1)(x+1)}$

Clues:

• Division flips to multiplication.

• What makes $(x-1)(x+1)$ when multiplied by $(x-1)$?

• Missing Answer: $\boxed{{\displaystyle \mathrm{<span\; style="color:\; \#cccccc;">x</span>}<span\; style="color:\; \#cccccc;">+</span>\mathrm{<span\; style="color:\; \#cccccc;">1</span>}}}$
  

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