Key Answer to Rectangular Coordinate system

Thursday, May 15, 2025

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Plot Like a Pro: Master the Rectangular Coordinate System in Just 5 Steps!

 Understanding Rectangular Coordinate Systems Basics

1. Introduction 

• Imagine this: You're locked in an intense game of Battleship, heart pounding as you call out *"B-5!"* Your opponent hesitates... then groans—direct hit! 🎯 What’s your secret? You’re not just guessing—you’re using math to dominate the grid.

  It turns out that Battleship is really a stealthy lesson in the rectangular coordinate system—the same tool scientists use to map stars, engineers use to design cities, and even your phone uses to navigate. Master this, and you’ll not only crush your opponents in games but unlock the hidden math behind everything from GPS to video game design.

  Ready to turn coordinates into your superpower? Let’s dive in!

• Objective: By the end of this lesson, you will be able to:

  • Plot points accurately on the coordinate plane.

  • Identify the quadrant of a given point.

  • Calculate the distance between two points using the distance formula.

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2. Prerequisite Knowledge Check

Before diving in, make sure you understand:
✔ Number lines (positive and negative directions).
✔ Ordered pairs (e.g., (3, 4) means x=3, y=4).
✔ Basic operations (addition, subtraction, squaring).

Need a refresher?

• Number Lines Explained

• Ordered Pairs Practice

---

3. Core Concept Explanation

What is the Rectangular Coordinate System?

• A grid formed by two perpendicular number lines:

  • x-axis (horizontal)

  • y-axis (vertical)

• Origin (0,0): Where the axes intersect.

• Quadrants: Four sections labeled I (+,+), II (−,+), III (−,−), IV (+,−).

Common Mistakes to Avoid:

❌ Swapping x and y coordinates (e.g., writing (y, x) instead of (x, y)).
❌ Forgetting negative signs when plotting points (e.g., (−2, 3) vs. (2, 3)).
❌ Misidentifying quadrants (e.g., (3,−5) is in Quadrant IV, not II).

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4. Worked Examples (Gradual Release of Responsibility)

Example 1 (I Do - Teacher Modeling)

Problem: Plot the point (2, 3).
Solution:

1. Start at the origin (0,0).

2. Move 2 units right along the x-axis (since x=2 is positive).

3. From there, move 3 units up along the y-axis (since y=3 is positive).

4. Mark the point where you land.
   Conclusion: (2, 3) is in Quadrant I.

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Example 2 (We Do - Guided Practice)

Problem: Plot the point (−1, 4).
Steps Together:

1. Start at (0,0).

2. Move 1 unit left (x=−1).

3. Move 4 units up (y=4).

4. Where is this point located?
   Answer: (−1, 4) is in Quadrant II.

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Example 3 (You Do Together - Collaborative Practice)

Problem: Identify the quadrant of (−3, −5).
Group Discussion:

• Is x negative or positive? (Negative)

• Is y negative or positive? (Negative)

• Which quadrant has (−,−)?
  Answer: Quadrant III.

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Example 4 (You Do - Independent Practice)

Problem: Calculate the distance between (1, 2) and (4, 6).
Formula: Distance = √((x₂−x₁)² + (y₂−y₁)²).
Worked Solution:

1. Subtract x-coordinates: 4−1 = 3 → 3² = 9.

2. Subtract y-coordinates: 6−2 = 4 → 4² = 16.

3. Add results: 9 + 16 = 25.

4. Take the square root: √25 = 5.
   Final Answer: The distance is 5 units.

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Example 5 (Challenge Problem - Extended Thinking)

Problem: Find the distance between (−2, 4) and (3, −1).
Steps:

1. Subtract x-coordinates: 3−(−2) = 5 → 5² = 25.

2. Subtract y-coordinates: −1−4 = −5 → (−5)² = 25.

3. Add results: 25 + 25 = 50.

4. Take the square root: √50 = 5√2.
   Final Answer: The distance is 5√2 units.

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5. Practice Problems (With Solutions)

Easy: Plot (0, −3). What quadrant is it located in?
Medium: What is the distance between (5, 1) and (2, −3)?
Hard: If point A is (−4, 0) and point B is (0, 3), what is the distance between them?

*Easy: (0,−3) lies on the **y-axis** (not in any quadrant). - 

**Medium: Distance = √((2−5)² + (−3−1)²) = √(9 + 16) = **5 units**. - 

***Hard: Distance = √((0−(−4))² + (3−0)²) = √(16 + 9) = **5 units**. 

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6. Real-World Applications

• GPS Navigation: Uses coordinates to pinpoint locations.

• Video Games: Characters move based on (x,y) positions.

• Architecture: Blueprints rely on grid systems for precision.

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7. Summary & Key Takeaways

✔ Points are written as (x, y).
✔ Quadrants are labeled counterclockwise: I → II → III → IV.
✔ Distance formula: √((x₂−x₁)² + (y₂−y₁)²).

 "X comes before Y in the alphabet, just like in (x,y)!"

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8. Further Reading & Resources

📺 Video: Coordinate Plane
📖 Book: Graphing Notebook
🔗 WOW Math: Math 8

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9. Interactive Element

Quick Quiz:

1. What quadrant is (−7, −2) in? (Answer: III)

2. What is the distance between (0,0) and (6,8)? (Answer: 10 units)

____________________________________________________________________________

10. Problem set. Fill in the blanks. Given the problems with a partial solution.

🔢 Problem Set: Coordinate Plane Puzzle Fixers

(For small groups of 3-4 students)*

Problem 1: The Misplaced Point

Partial Solution:
*"The point (3, -2) was plotted in Quadrant II, but that’s incorrect because..."*
Task:

1. Identify the error.

2. Correctly plot and label the point.

3. Challenge: Find a point in Quadrant II with the same y-coordinate.

Self-Check: The corrected point forms a rectangle with (-3, -2), (3, -2), and (-3, 2).

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Problem 2: Distance Detective

Partial Solution:
*"The distance between (1, 4) and (5, 1) is calculated as:
√‾(5−1)² + (1−4)² = √‾16 + ___ = √‾___ = 5 units."*
Task:

1. Fill in the missing steps.

2. Challenge: Find a point on the y-axis that’s exactly 5 units from (1, 4).

Hint: The missing term is a perfect square.

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Problem 3: Reflection Riddle

Partial Solution:
*"When (2, -3) is reflected over the x-axis, the new point is (2, 3). If you then reflect it over the y-axis, the final point is (___ , ___)."*
Task:

1. Complete the transformation.

2. Challenge: What single reflection would take (2, -3) directly to the final point?

Visual Aid: Provide small mirrors to test predictions.

---

Problem 4: Battleship Logic

Partial Solution:
*"A ship stretches from (-1, 2) to (-1, 5). Its length is 3 units. If another ship runs from (3, -4) to (___, -4) and is twice as long, the missing x-coordinate is ___."*
Task:

1. Find the endpoint.

2. Challenge: Could both ships be sunk by hitting (-1, 4) and (5, -4)? Explain.

Self-Check: The completed ship covers 3 integer x-values.

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Problem 5: Midpoint Mystery

Partial Solution:
*"The midpoint between (-6, 3) and (2, -1) is:
( (-6 + ___)/2 , (3 + )/2 ) = ( , ___)."*
Task:

1. Fill in the blanks.

2. Challenge: Find the endpoint if (-2, 1) is the midpoint and the other endpoint is (0, 5).

Hint: The missing midpoint coordinates are integers.

---

Problem 6: Shape 

Partial Solution:
"Points A(1, 1), B(1, 4), and C(5, 1) form a right triangle because the slopes of AB and AC are ___ and ___. The area is ___ square units."
Task:

1. Calculate the missing slopes and area.

2. Challenge: Add point D to make a rectangle.

Clue: Area = ½ × base × height.

📝 Practice Problems: Rectangular Coordinate System

Directions: Solve each problem with your group. Show all work and justify your answers.

Basic Skills (Problems 1-3)

1. Plotting Points
   Plot and label these points: A(2, 5), B(-3, 0), C(-1, -4). Identify the quadrant or axis for each.

2. Distance Warm-Up
   Find the distance between (0, 0) and (6, 8).

3. Missing Coordinate
   If (4, y) is 5 units from (1, 2), find *y*. (Hint: There are two possible answers!)

Mid-Level Challenges (Problems 4-6)

4. Quadrant Logic
   "Point P has a negative x-coordinate and a positive y-coordinate. If you reflect P over the y-axis, which quadrant is the new point in?"

5. Perimeter Puzzle
   Three vertices of a rectangle are at (1, 1), (1, 4), and (5, 4). Find:
   a) The fourth vertex.
   b) The perimeter.

6. Collinear Points
   Show whether (2, 3), (4, 6), and (0, 0) lie on the same straight line.

Advanced Applications (Problems 7-10)

7. Battleship Midpoint
   A ship’s midpoint is at (3, 2). One endpoint is (5, 5). Find the other endpoint.

8. Circle Equation
   A circle’s center is at (-1, 3) and passes through (2, 7). Find its radius.

9. Slope Analysis
   A line passes through (-2, 4) and (1, -2). Find:
   a) The slope.
   b) The y-intercept.

10. Real-World GPS
    You’re at (3, -1) and need to reach a café 10 units away at (x, 5). Find *x*.

Worksheet PDF Download

_________________________________________________________________________

🔑 Solutions

1. Plotting Points

   • A(2,5): Quadrant I

   • B(-3,0): On x-axis

   • C(-1,-4): Quadrant III

2. Distance
   √‾(6−0)² + (8−0)² = √‾36 + 64 = √‾100 = 10 units

3. Missing Coordinate
   √‾(4−1)² + (y−2)² = 5 → 9 + (y−2)² = 25 → *y* = 6 or -2

4. Reflection
   Original: Quadrant II → Reflected: Quadrant I

5. Rectangle
   a) Fourth vertex: (5, 1)
   b) Perimeter: 2(4 + 3) = 14 units

6. Collinear Check
   Slope (0,0)→(2,3) = 1.5; Slope (2,3)→(4,6) = 1.5 → Yes, collinear

7. Endpoint
   Midpoint formula → Other endpoint: (1, -1)

8. Radius
   √‾(2−(−1))² + (7−3)² = √‾9 + 16 = 5 units

9. Line Equation
   a) Slope = (−2−4)/(1−(−2)) = -2
   b) y = -2x + 0 → y-intercept: (0, 0)

10. GPS Café
    √‾(x−3)² + (5−(−1))² = 10 → (x−3)² + 36 = 100 → *x* = 11 or -5

_________________________________________________________________________

SIMPLIFIED TIPS FOR LEARNING.

🌟 5 Easy Tips to Master the Rectangular Coordinate System

1. Turn Coordinates into a Game 🎲

• Battleship Method: Play the classic game Battleship (or use graph paper) to practice plotting points like (3,5) and (-2,4).

• Treasure Hunts: Hide "treasure" in your house and describe its location using coordinates (e.g., "3 steps right from the door, 2 steps up").

2. Remember the "X to the Sky, Y to the Thigh" Trick ☝️

• X-axis: Think of it as the ground (left/right).

• Y-axis: Imagine it as a ladder (up/down).

• Quadrants: Use the sign rules:

  • Quadrant I (+, +): Happy (both positive)

  • Quadrant II (−, +): Sad left, happy up

  • Quadrant III (−, −): Sad all around

  • Quadrant IV (+, −): Happy right, sad down

3. Use Real-Life Examples 

• GPS: Google Maps uses coordinates to find locations.

• Sports: Basketball court positions (e.g., (0,0) = center court).

• Art: Pixel art and video game design rely on grids.

4. Master Formulas with Simple Stories 📖

• Distance Formula: Imagine a right triangle between two points. Use the Pythagorean Theorem:

  • "Walk the x-distance (base), climb the y-distance (height), then find the hypotenuse (distance)."
    Example: Distance between (1,2) and (4,6) = √((4-1)² + (6-2)²) = 5 units.

• Midpoint Formula: Think of it as averaging two locations:
  "Add the x’s, divide by 2. Add the y’s, divide by 2. Boom—middle point!"

5. Practice with Mini-Challenges ✏️

• Daily Drills: Plot 3 random points every day and name their quadrants.

• Flashcards: Write point pairs on one side and their distance/midpoint on the back.

• Error Analysis: Intentionally make mistakes (e.g., swap x/y) and correct them.

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🎯 Pro Tip:

Struggling with negatives? Use colored pens:

• 🔴 Red for negative coordinates

• 🔵 Blue for positive coordinates

Example: (-3, 5) = 3 red steps left, 5 blue steps up.

_______________________________________________________________________

Related Links:

BYJU'S

Desmos

CueMath

Downloadable Practice Problems.

Solutions to Practice Problems

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Plot Like a Pro: Master the Rectangular Coordinate System

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Operations on Rational Expressions Simplified

Saturday, May 10, 2025

 Operations on Rational Expressions

Photo Credit

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.

---

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).

---

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 $\frac{1}{x+1}$).

---

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)

---

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 $x\mathrm{\ne }1$.

---

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:

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

---

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 $h$:

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

_____________________________________________________________________

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

Click the PDF Worksheet Download

Click: Answer Keys

Post your answers below and discuss! 🚀

Related Links

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Rational Expressions | A Comprehensive Guide

Thursday, May 8, 2025

 Rational Expressions: A Comprehensive Guide

1. Introduction 

Imagine you’re baking cookies and need to adjust a recipe. The original recipe calls for (2/x) cups of sugar per batch, but you want to make (x/(x+1)) batches. How much sugar do you need in total?

Objective:

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

• Simplify, multiply, divide, add, and subtract rational expressions.

• Solve real-world problems involving rational expressions.

---

Prerequisite Knowledge Check

Before we start, make sure you know:

1. Factoring polynomials [e.g., $x^2-4=(x+2)(x-2)$]

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

3. Finding the Least Common Denominator (LCD) [e.g., LCD of $\frac{1}{x}$ and $\frac{1}{x+1}$ is $x(x+1)$]

---

Core Concept Explanation (I Do – Teacher Models)

What is a Rational Expression?

A rational expression is a fraction where the numerator and denominator are polynomials.

Examples:

• $\frac{3x}{x+2}$

• $\frac{x^2-4}{x^2+5x+6}$

Simplifying Rational Expressions

Step 1: Factor numerator and denominator.

Step 2: Cancel common factors.

Example: Simplify $\frac{x^2-9}{x^2+6x+9}$

1. Factor:

   $$\frac{(x+3)(x-3)}{(x+3{)}^{\mathrm{2}}}$$

2. Cancel $(x+3)$:

   $$\frac{x-3}{x+3}$$
   

---

Guided Practice (We Do – Teacher & Students Together)

Problem: Simplify $\frac{2x^2-8}{x^2-x-6}$

Teacher’s Guidance:

• Question: What’s the first step? (Factor numerator and denominator.)

• Question: What factoring techniques apply here? (Difference of squares, trinomial factoring.)

Solution:

1. Factor numerator: $2x^2-8=2(x^2-4)=2(x+2)(x-2)$
   

2. Factor denominator: $x^2-x-6=(x-3)(x+2)$
   

3. Rewrite:

   $$\frac{2(x+2)(x-2)}{(x-3)(x+2)}$$

4. Cancel $(x+2)$:

   $$\frac{2(x-2)}{x-\mathrm{3}}$$

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Independent Practice (You Do – Students Try Alone)

Try these problems on your own before checking the solutions!

1. Simplify $\frac{3x^2-12}{x^2+4x+4}$
   (Hint: Factor the numerator and denominator first.)

2. Simplify $\frac{x^2-5x+6}{x^2-9}$

Solutions:

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

2. $\frac{x-2}{x+3}$

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Common Mistakes & Troubleshooting

🚨 Watch out for these errors!

• Forgetting to factor completely (e.g., missing common factors).

• Canceling terms, not factors (e.g., $\frac{x+3}{x}$ cannot cancel $x$).

• Ignoring restrictions (denominator cannot be zero).

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Real-World Application

Example: Engineers use rational expressions to calculate resistance in parallel circuits:

$$\frac{1}{R_{total}}=\frac{1}{R_1}+\frac{1}{R_{\mathrm{2}}}$$

Simplifying helps find the total resistance efficiently!

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Summary & Key Takeaways

✔ A rational expression is a fraction of polynomials.
✔ Simplify by factoring and canceling common factors.
✔ Never divide by zero—always state excluded values.

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Practice & Extension

Extra Practice:

1. Simplify $\frac{4x^2-16}{2x^2-8x+8}$

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

Challenge Question:
Solve for $x$:

$$\frac{x}{x-2}-\frac{3}{x+2}=\frac{8}{x^2-\mathrm{4}}$$

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Further Resources

📺 Video Tutorials: Inorganic Chemistry Tutor - Simplifying Rational Expressions
📖 Interactive Practice: Desmos Rational Simplifier

Next Lesson: Solving Rational Equations

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More Illustrative Illustrations.

Example 1: Simplify

Expression: $\frac{x^2-4}{x^2+4x+4}$

Solution:

Numerator: $x^2-4=(x+2)(x-2)$

Denominator: $x^2+4x+4=(x+2{)}^2$

Simplified: $\frac{(x+2)(x-2)}{(x+2{)}^2}=\frac{x-2}{x+2}$

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Example 2: Multiply

Expression: $\frac{x+3}{x^2-9}\times \frac{x-3}{x+1}$

Solution:

Factor denominators: $x^2-9=(x+3)(x-3)$

Multiply numerators: $(x+3)(x-3)=x^2-9$

Multiply denominators: $(x+3)(x-3)(x+1)$

Simplified: $\frac{x^2-9}{(x+3)(x-3)(x+1)}=\frac{1}{x+1}$

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Example 3: Divide

Expression: $\frac{2x}{x^2-1}÷\frac{4x^2}{x+1}$

Solution:

Reciprocal and multiply: $\frac{2x}{x^2-1}\times \frac{x+1}{4x^2}$

Factor denominator: $x^2-1=(x+1)(x-1)$

Simplified: $\frac{2x(x+1)}{(x+1)(x-1)(4x^2)}=\frac{1}{2x(x-1)}$

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Example 4: Add

Expression: $\frac{1}{x}+\frac{2}{x+1}$

Solution:

Common denominator: $x(x+1)$

Rewrite fractions: $\frac{x+1}{x(x+1)}+\frac{2x}{x(x+1)}$

Combine numerators: $\frac{x+1+2x}{x(x+1)}=\frac{3x+1}{x(x+1)}$

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Example 5: Subtract

Expression: $\frac{5}{x-2}-\frac{3}{x+2}$

Solution:

Common denominator: $(x-2)(x+2)$

Rewrite fractions: $\frac{5(x+2)}{(x-2)(x+2)}-\frac{3(x-2)}{(x-2)(x+2)}$

Combine numerators: $\frac{5x+10-3x+6}{(x-2)(x+2)}=\frac{2x+16}{(x-2)(x+2)}$

More examples with Partial Solutions. Fill in the blanks what is/are the missing part(s) of the solution.

1. Simplify

Expression: $\frac{x^2-9}{x^2+6x+9}$

Steps:

Numerator:

Denominator: $x^2+6x+9=(\underset{‾}{}{)}^2$

Simplified: ​​​

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2. Multiply

Expression: $\frac{x+2}{x^2-4}\times \frac{x-2}{x+1}$

Steps:

Factor denominator:

Multiply numerators: $(x+2)(x-2)= \underset{}{<a\; href="https://www.blogger.com/comment/fullpage/post/9093914164408134262/7268964984793473683"\; target="\_blank">\_\_\_\_\_\_\_\_\_</a>}$

Result: _________

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3. Divide

Expression: $\frac{3x}{x^2-16}÷\frac{6x^2}{x+4}$

Steps:

Reciprocal: $\frac{3x}{x^2-16}\times$

Factor: _________

Simplified form: _________

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4. Add

Expression: $\frac{2}{x}+\frac{3}{x+2}$

Steps:

Common denominator: ​

Combine: $\frac{2(\underset{‾}{})+3(\underset{‾}{})}{x(x+2)}$

Result: $\frac{\underset{‾}{}}{x(x+2)}$

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5. Subtract

Expression: $\frac{4}{x-3}-\frac{2}{x+1}$

Steps:

Common denominator: ___________​

Combine: $\frac{4(\underset{‾}{})-2(\underset{‾}{})}{(x-3)(x+1)}$

Result: $\frac{\underset{‾}{}}{(x-3)(x+1)}$

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6. Complex Fraction

Expression: $\frac{\frac{x}{x+1\mathrm{<br>}}}{\frac{\mathrm{<br>}\mathrm{<br>}2x}{x^2-1}}$

Steps:

Rewrite: $\frac{x}{x+1}\times \frac{\underset{‾}{}}{\underset{‾}{}}$

Factor:

Simplified: $\frac{\underset{‾}{}}{2}$

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7. Find Restrictions

Expression: $\frac{x^2+5x+6}{x^2-4}$

Steps:

Numerator:

Denominator:

Excluded values: $x\mathrm{\ne \; <a\; href="https://www.blogger.com/comment/fullpage/post/9093914164408134262/7268964984793473683"\; target="\_blank">\_\_\_\_\_\_\_\_\_\_</a>\; and }$$x\mathrm{\ne <a\; href="https://www.blogger.com/comment/fullpage/post/9093914164408134262/7268964984793473683"\; target="\_blank">\_\_\_\_\_\_\_\_\_\_</a>}$

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8. Multiply and Simplify

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

Steps:

Factor:

Cancel: $\frac{(x+1)\cancel{(x-1)}\cancel{(x+2)}}{\cancel{(x+2)}\cancel{(x-1)}}$

Result:__________​

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9. Add and Simplify

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

Steps:

Factor: 

Common denominator: _________
$\underline{}$
Result: $\frac{\underset{‾}{}}{(x-1)(x+1)}$

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10. Subtract and Simplify

Expression: $\frac{5}{x^2-x}-\frac{3}{x-1}$

Steps:

Factor:

Common denominator:___________​

Result: $\frac{\underset{‾}{}}{x(x-1)}$

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Answer key

1. $\frac{x-3}{x+3}$

2. $\frac{1}{x+1}$

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

4. $\frac{5x+4}{x(x+2)}$

5. $\frac{2x+10}{(x-3)(x+1)}$

6. $\frac{x-1}{2}$

7. $x\mathrm{\ne }2,x\mathrm{\ne }-2$
   

8. $x+1$
   

9. $\frac{2}{x-1}$

10. $\frac{}{}$−3x+5x(x−1)​

_________________________________________________

Real-World Rational Expression Problems with Answers

1. Fuel Efficiency

   A car travels $\frac{300}{x+5}$ miles using $\frac{15}{x}$ gallons of gas.
   Simplify the mpg expression.
   Solution: $\frac{20x}{x+5}$ , *(x ≠ 0, -5)*

2. Construction Costs

   Building $\frac{x^2-25}{x-5}$ shelves costs $600.
   Find the cost per shelf.
   Solution: $\frac{600}{x+5}$ (x ≠ 5)

3. Baking Recipe

   A cake needs $\frac{3x}{4}$ cups of flour and $\frac{x}{2}$ cups of sugar.
   Simplify: Flour and sugar ratio.
   Solution: $\frac{3}{2}$ (x ≠ 0)

4. Train Speed

   A train covers $\frac{x^2-36}{x-6}$ km in $\frac{x+6}{3}$ hours.
   Find speed in km/h.
   Solution: 3 (x ≠ 6)

5. Medicine Dosage

   Child's dose is $\frac{4x}{x^2-16}$ mL (adult dose = x mL).
   Simplify dosage.
   Solution: $\frac{4}{x+4}$ , (x ≠ 4, -4)

6. Garden Soil

   Mix $\frac{x+3}{x^2-9}$ lbs of compost with $\frac{1}{x-3}$ lbs of soil.
   Total weight?
   Solution: $\frac{2}{x-3}$ (x ≠ 3, -3)

7. Electricity Bill

   $\frac{x^2-49}{x+7}$ kWh costs $84.
   Cost per kWh?
   Solution: $\frac{84}{x-7}$, (x ≠ -7)

8. Paint Coverage

   $\frac{6x}{x^2-9}$ gallons paints $\frac{x+3}{2}$ walls.
   Gallons per wall?
   Solution: $\frac{12}{x-3}$, (x ≠ 3, -3)

9. Investment Growth

   $$\frac{x-5}{x^2-25}$ grows to $$\frac{1}{x+5}$ in 1 year.
   Growth factor?
   Solution: 1 , (x ≠ 5, -5)

10. Chemistry Lab

    $\frac{x^2-4x+4}{x-2}$ mL acid mixed with $\frac{2x}{5}$ mL water.
    Acid: water ratio?
    Solution: $\frac{5(x-2)}{2x}$ (x ≠ 0, 2)

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Detailed Solution.

​Complete Rational Expressions Practice Worksheet 

1. Fuel Efficiency

Problem:
A car travels $\frac{300}{x+5}$ miles using $\frac{15}{x}$ gallons of gas. Simplify the mpg expression.

Solution:

$$\frac{\frac{300}{x+5}}{\frac{15}{x}}=\frac{300}{x+5}\times \frac{x}{15}=\frac{300x}{\mathrm{15 }(\mathrm{x }+5) }= \frac{20x}{x+5}$$

Restrictions: $x\mathrm{\ne }0,-5$

Answer: $\boxed{{\displaystyle \frac{20x}{x+5}}}$

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2. Construction Costs

Problem:
Building $\frac{x^2-25}{x-5}$ shelves costs $600. Find the cost per shelf.

Solution:

$$\frac{x^2-25}{x-5}=\frac{(x+5)(x-5)}{x-5}=x+5(x\mathrm{\ne }5)$$
$$\text{Cost per shelf}=\frac{600}{x+\mathrm{5}}$$

Answer: $\boxed{{\displaystyle \frac{600}{x+5}}}$

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3. Baking Ratio

Problem:
A recipe uses $\frac{3x}{4}$ cups of flour and $\frac{x}{2}$ cups of sugar. Simplify the flour-to-sugar ratio.

Solution:

$$\frac{\frac{3x}{4}}{\frac{x}{2}}=\frac{3x}{4}\times \frac{2}{x}=\frac{6x}{4x}=\frac{3}{2}$$

Restrictions: $x\mathrm{\ne }0$
Answer: $\boxed{{\displaystyle \frac{3}{2}}}$

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4. Speed Calculation

Problem:
A train covers $\frac{x^2-36}{x-6}$ km in $\frac{x+6}{3}$ hours. Find its speed (km/h).

Solution:

$$\frac{x^2-36}{x-6}=\frac{(x+6)(x-6)}{x-6}=x+6(x\mathrm{\ne }6)$$
$$\text{Speed}=\frac{x+6}{\frac{x+6}{3}}=3$$

Answer: $\boxed{{\displaystyle 3}}$

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5. Medicine Dosage

Problem:
A child’s dose is $\frac{4x}{x^2-16}$ mL (adult dose = $x$ mL). Simplify the expression.

Solution:

$$\frac{4x}{(x+4)(x-4)}=\frac{4}{x+4}(x\mathrm{\ne }4,-4)$$

Answer: $\boxed{{\displaystyle \frac{4}{x+4}}}$

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6. Gardening Mixture

Problem:
Mix $\frac{x+3}{x^2-9}$ lbs of compost with $\frac{1}{x-3}$ lbs of soil. Find the total weight.

Solution:

$$\frac{x+3}{(x+3)(x-3)}+\frac{1}{x-3}=\frac{2}{x-3}(x\mathrm{\ne }3,-3)$$

Answer: $\boxed{{\displaystyle \frac{2}{x-3}}}$

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7. Electricity Cost

Problem:
$\frac{x^2-49}{x+7}$ kWh costs $84. Find the cost per kWh.

Solution:

$$\frac{x^2-49}{x+7}=x-7(x\mathrm{\ne }-7)$$
$$\text{Cost per kWh}=\frac{84}{x-\mathrm{7}}$$

Answer: $\boxed{{\displaystyle \frac{84}{x-7}}}$

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8. Paint Coverage

Problem:
$\frac{6x}{x^2-9}$ gallons covers $\frac{x+3}{2}$ walls. Find gallons per wall.

Solution:

$$\frac{6x}{(x+3)(x-3)}÷\frac{x+3}{2}=\frac{12x}{(x+3{)}^2(x-3)}$$

Restrictions: $x\mathrm{\ne }3,-3$
Answer: $\boxed{{\displaystyle \frac{12}{x^2-9}}}$

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9. Investment Growth

Problem:
An investment grows from $\frac{x-5}{x^2-25}$ to $\frac{1}{x+5}$ dollars. Find the growth factor.

Solution:

$$\frac{x-5}{(x+5)(x-5)}=\frac{1}{x+5}(x\mathrm{\ne }5)$$
$$\text{<br>}$$

Answer: $\boxed{{\displaystyle 1}}$ (no growth)

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10. Chemical Solution Ratio

Problem:
A lab has $\frac{x^2-4x+4}{x-2}$ mL Solution A and $\frac{2x}{5}$ mL Solution B. Find the A : B ratio.

Solution:

$$\frac{(x-2{)}^2}{x-2}=x-2(x\mathrm{\ne }2)<br>$$$$\text{<br>}\text{Ratio}=\frac{x-2}{\frac{2x}{5}}=\frac{5(x-2)}{2x}$$

Answer: $\boxed{{\displaystyle \frac{5(x-2)}{2x}}}$

_________________________________________________

Exercises:

Simplify each given problem. The answers are provided with this symbol (→). Just show how the answer ends that way.

1. $\frac{\mathrm{<b>3</b>}\mathrm{<b>x</b>}}{\mathrm{<b>6</b>}{\mathrm{<b>x</b>}}^{\mathrm{<b>2</b>}}}$
   → $\frac{\mathrm{<span\; style="font-size:\; small;">1</span>}}{\mathrm{<span\; style="font-size:\; small;">2</span>}\mathrm{<span\; style="font-size:\; small;">x</span>}}$ (x ≠ 0)

2. $\frac{{\mathrm{<b>x</b>}}^{\mathrm{<b>2</b>}}<b>-</b>\mathrm{<b>4</b>}}{\mathrm{<b>x</b>}<b>+</b>\mathrm{<b>2</b>}}$
   → $\frac{x-2}{1}$ = x-2, (x ≠ -2)

3. $\frac{\mathrm{<b>5</b>}\mathrm{<b>x</b>}<b>+</b>\mathrm{<b>10</b>}}{\mathrm{<b>15</b>}\mathrm{<b>x</b>}}$
   → $\frac{x+2}{3x}$, (x ≠ 0)

4. $\frac{{\mathrm{<b>x</b>}}^{\mathrm{<b>2</b>}}<b>-</b>\mathrm{<b>9</b>}}{{\mathrm{<b>x</b>}}^{\mathrm{<b>2</b>}}<b>+</b>\mathrm{<b>6</b>}\mathrm{<b>x</b>}<b>+</b>\mathrm{<b>9</b>}}$
   → $\frac{x-3}{x+3}$, (x ≠ -3)

5. $\frac{\mathrm{<b>2</b>}{\mathrm{<b>x</b>}}^{\mathrm{<b>2</b>}}<b>-</b>\mathrm{<b>8</b>}\mathrm{<b>x</b>}}{\mathrm{<b>4</b>}\mathrm{<b>x</b>}}$
   → $\frac{x-2}{2}$(x ≠ 0)

6. $\frac{{\mathrm{<b>x</b>}}^{\mathrm{<b>2</b>}}<b>-</b>\mathrm{<b>5</b>}\mathrm{<b>x</b>}<b>+</b>\mathrm{<b>6</b>}}{{\mathrm{<b>x</b>}}^{\mathrm{<b>2</b>}}<b>-</b>\mathrm{<b>9</b>}}$
   → $\frac{x-2}{x+3}$ (x ≠ ± 3)

7. $\frac{\mathrm{<b>3</b>}{\mathrm{<b>x</b>}}^{\mathrm{<b>3</b>}}}{\mathrm{<b>9</b>}{\mathrm{<b>x</b>}}^{\mathrm{<b>2</b>}}<b>+</b>\mathrm{<b>6</b>}\mathrm{<b>x</b>}}$
   → $\frac{x^2}{3x+2}$ (x ≠ 0,-⅔)

8. $\frac{{\mathrm{<b>x</b>}}^{\mathrm{<b>2</b>}}<b>-</b>\mathrm{<b>16</b>}}{{\mathrm{<b>x</b>}}^{\mathrm{<b>2</b>}}<b>-</b>\mathrm{<b>8</b>}\mathrm{<b>x</b>}<b>+</b>\mathrm{<b>16</b>}}$
   → $\frac{x+4}{x-4}$ (x ≠ 4)

9. $\frac{\mathrm{<b>6</b>}{\mathrm{<b>x</b>}}^{\mathrm{<b>2</b>}}\mathrm{<b>y</b>}}{\mathrm{<b>12</b>}\mathrm{<b>x</b>}{\mathrm{<b>y</b>}}^{\mathrm{<b>2</b>}}}$
   → $\frac{x}{2y}$ , (x ≠ 0, y ≠ 0)

10. $\frac{{\mathrm{<b>x</b>}}^{\mathrm{<b>2</b>}}<b>+</b>\mathrm{<b>x</b>}<b>-</b>\mathrm{<b>6</b>}}{{\mathrm{<b>x</b>}}^{\mathrm{<b>2</b>}}<b>-</b>\mathrm{<b>4</b>}\mathrm{<b>x</b>}<b>+</b>\mathrm{<b>4</b>}}$
    → $\frac{\mathrm{<span\; style="font-size:\; x-small;">x</span>}<span\; style="font-size:\; x-small;">+</span>\mathrm{<span\; style="font-size:\; x-small;">3</span>}}{\mathrm{<span\; style="font-size:\; x-small;">x</span>}<span\; style="font-size:\; x-small;">-</span>\mathrm{<span\; style="font-size:\; x-small;">2</span>}}$ , (x ≠ 2)

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Worksheet Answer Key

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Inorganic Chemistry Tutor

Desmos Rational Simplifier

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