Extraction Lab Worksheet

REF: R-REALM-01

Extraction Lab Report

Operator: ____________________

Date: __________

Subject: Simplifying & Operating on Rational Expressions

Phase 01: Factor & Simplify

Factor the numerator and denominator completely. Extract (cancel) the common factors to simplify the expression.

1.1

\[ \frac{8x^3}{24x^5} \]

1.2

\[ \frac{x^2 - 16}{x - 4} \]

1.3

\[ \frac{x^2 + 7x + 10}{x^2 - 25} \]

1.4

\[ \frac{2x^2 - 8x}{x^2 - 5x + 4} \]

Phase 02: Multiplying (Smash & Slash)

2.1

\[ \frac{x^2 - 1}{x + 3} \cdot \frac{x^2 + 5x + 6}{x - 1} \]

2.2

\[ \frac{5x}{x^2 - 4} \cdot \frac{x+2}{10x^2} \]

REF: R-REALM-01

Phase 03: Dividing (Keep-Change-Flip)

3.1

\[ \frac{x^2 - 9}{2x} \div \frac{x + 3}{4x^2} \]

3.2

\[ \frac{x^2 + 4x + 4}{x^2 - 4} \div (x + 2) \]

Phase 04: Illegal Maneuver Detection

Analyze the "solution" below. Identify the error made and explain why it is mathematically illegal.

System Log #404:

\[ \frac{x + 7}{x + 14} = \frac{\cancel{x} + 7}{\cancel{x} + 14} = \frac{7}{14} = \frac{1}{2} \]

Error Identification:

Correct Strategy:

Extraction Lab Answer Key

KEY: R-REALM-01

Lab Report Answer Key

Simplifying, Multiplying & Dividing Rational Expressions

Phase 01: Simplifying

1.1 Result:

\[ \frac{1}{3x^2} \]

Factors: \(\frac{8x^3}{3 \cdot 8x^3 \cdot x^2}\)

1.2 Result:

\[ x + 4 \]

Factors: \(\frac{(x-4)(x+4)}{x-4}\)

1.3 Result:

\[ \frac{x + 2}{x - 5} \]

Factors: \(\frac{(x+2)(x+5)}{(x-5)(x+5)}\)

1.4 Result:

\[ \frac{2x}{x - 1} \]

Factors: \(\frac{2x(x-4)}{(x-4)(x-1)}\)

Phase 02: Multiplying

2.1 Result:

\[ (x + 1)(x + 2) \text{ or } x^2 + 3x + 2 \]

Factored Form: \(\frac{(x-1)(x+1)}{x+3} \cdot \frac{(x+2)(x+3)}{x-1}\)

2.2 Result:

\[ \frac{1}{2x(x - 2)} \]

Factored Form: \(\frac{5x}{(x-2)(x+2)} \cdot \frac{x+2}{2 \cdot 5x \cdot x}\)

Phase 03: Dividing

3.1 Result:

\[ 2x(x - 3) \text{ or } 2x^2 - 6x \]

Process: \(\frac{(x-3)(x+3)}{2x} \cdot \frac{4x^2}{x+3}\)

3.2 Result:

\[ \frac{1}{x - 2} \]

Process: \(\frac{(x+2)^2}{(x-2)(x+2)} \cdot \frac{1}{x+2}\)

Phase 04: Error Detection

Error Identification:

The error is "canceling terms" rather than factors. In the expression \(\frac{x+7}{x+14}\), the \(x\)'s are addends (parts of a sum). You can only divide out (cancel) values that are being multiplied to the entire numerator and denominator.

Correct Strategy:

Determine if the numerator and denominator can be factored. Since neither \(x+7\) nor \(x+14\) can be factored further, and they are not identical, the expression is already in its simplest form. It cannot be simplified to \(\frac{1}{2}\).

Common Ground Slides

Common Ground

Rational Realms // Lesson 1.2

Operation: Addition & Subtraction

The Denominator Law

Just like regular fractions, you cannot combine rational expressions unless they have the same foundation.

\[ \frac{A}{C} \pm \frac{B}{C} = \frac{A \pm B}{C} \ bond \]

Inspection Check:

• If denominators match: Just add/sub the numerators.
• If they DON'T match: We must build an LCD.
• The denominator stays the same in the final step!

Construction Protocol

Step 1: Build the LCD

LCD Definition

The Least Common Denominator is the smallest expression that contains every factor from both denominators.

"Collect the unique factors like building blocks. Don't double count if they are already there!"

LCD Example:

Denom 1: \( (x+1)(x+2) \)

Denom 2: \( (x+2)(x+3) \)

The LCD is:

\( (x+1)(x+2)(x+3) \)

THE SUBTRACTION TRAP

When subtracting, the negative sign must distribute to every term in the second numerator.

\( - (x + 5) = -x - 5 \)

🚨 DEADLY ERROR 🚨

\[ \frac{5}{x} - \frac{x+2}{x} = \frac{5-x+2}{x} \]

Forgot to distribute to the +2!

Full Site Plan

1

Factor

Factor both denominators completely.

2

Build LCD

Identify all unique factors needed.

3

Upgrade

Multiply numerators by "what's missing."

4

Combine

Add/Sub numerators. Keep LCD.

On-Site Test

Build the common ground:

\[ \frac{3}{x+2} + \frac{x}{x-1} \]

LCD \( (x+2)(x-1) \)

Numerator \( 3(x-1) + x(x+2) \)

Foundation Check Worksheet

SITE: C-GROUND-02

Foundation Check

Worker Name: ____________________

Date: __________

Subject: Adding and Subtracting Rational Expressions

Phase 01: LCD Blueprint

Identify the Least Common Denominator (LCD) for each pair of expressions. Factor first if necessary!

1.1

\( \frac{1}{6x} \) and \( \frac{1}{8x^2} \)

LCD:

1.2

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

LCD:

1.3

\( \frac{1}{x^2 - 4} \) and \( \frac{1}{x+2} \)

LCD:

1.4

\( \frac{1}{x^2 + 5x + 6} \) and \( \frac{1}{x^2 - 4} \)

LCD:

Phase 02: Level Foundation (Same Denom)

2.1

\[ \frac{5x + 1}{x + 5} + \frac{x + 29}{x + 5} \]

2.2

\[ \frac{x^2 + 3x}{x - 4} - \frac{28}{x - 4} \]

SITE: C-GROUND-02

Phase 03: Building Up (Different Denoms)

3.1

\[ \frac{4}{x + 2} + \frac{3}{x} \]

3.2

\[ \frac{x}{x - 3} - \frac{2}{x + 3} \]

3.3

\[ \frac{1}{x^2 - 1} + \frac{1}{x + 1} \]

Safety Inspection

Detected: The Subtraction Trap

A worker submitted the following calculation for \( \frac{8}{x} - \frac{x+5}{x} \):

"The answer is \( \frac{8-x+5}{x} \)"

What went wrong?

Correct Final Expression:

Foundation Check Answer Key

KEY: C-GROUND-02

Foundation Check Answer Key

Addition & Subtraction of Rational Expressions

Phase 01: LCD Blueprint

1.1 LCD:

\( 24x^2 \)

1.2 LCD:

\( (x+3)(x-2) \)

1.3 LCD:

\( (x-2)(x+2) \)

1.4 LCD:

\( (x+3)(x+2)(x-2) \)

Phase 02: Level Foundation

2.1 Result:

\[ \frac{6x + 30}{x + 5} = \frac{6(x+5)}{x+5} = 6 \]

Note: Expression simplifies perfectly!

2.2 Result:

\[ \frac{x^2 + 3x - 28}{x - 4} = \frac{(x+7)(x-4)}{x-4} = x + 7 \]

Note: Factoring the numerator is essential here.

Phase 03: Building Up

3.1 Result:

\[ \frac{4x + 3(x+2)}{x(x+2)} = \frac{7x+6}{x(x+2)} \]

3.2 Result:

\[ \frac{x(x+3) - 2(x-3)}{(x-3)(x+3)} = \frac{x^2+x+6}{(x-3)(x+3)} \]

3.3 Result:

\[ \frac{1 + 1(x-1)}{(x-1)(x+1)} = \frac{x}{(x-1)(x+1)} \]

Phase 04: Inspection Report

What went wrong?

The worker failed to distribute the negative sign to the entire second numerator. When subtracting \(\frac{x+5}{x}\), the negative applies to both the \(x\) and the \(5\).

Correct Final Expression:

\[ \frac{8 - (x + 5)}{x} = \frac{8 - x - 5}{x} = \frac{3 - x}{x} \]

Equation Escape Slides

Equation Escape

Rational Realms // Lesson 1.3

System Status: Locked

The "Clear" Protocol

To escape fractions, we must destroy the denominators.

Master Key:

Multiply EVERY TERM by the Least Common Denominator (LCD).

1

The LCD cancels all denominators.

2

The equation becomes a normal linear or quadratic equation.

3

Solve for the variable as usual.

System Override Case

LOG: OVERRIDE-01

Equation:

\[ \frac{3}{x} + \frac{1}{2} = \frac{5}{x} \]

Identify LCD:

2x

1

\[ \mathbf{2x} \left( \frac{3}{x} \right) + \mathbf{2x} \left( \frac{1}{2} \right) = \mathbf{2x} \left( \frac{5}{x} \right) \]

2

\[ 6 + x = 10 \]

3

\[ x = 4 \]

SECURITY ALERT:

EXTRANEOUS SOLUTIONS

Sometimes your "answer" is a virus.

If your final solution makes ANY original denominator equal zero, you must delete it.

"The math worked, but the system crashed."

Threat Scenario:

\[ \frac{x}{x - 5} = \frac{5}{x - 5} \]

Solving gives...

x = 5

INVALID: Denominator becomes 0!

Real-World Relay

The Work Rate:

If an object takes T time to finish a job, its rate is:

\[ \frac{1}{T} \]

The Team Equation:

\[ \frac{1}{\text{Part A}} + \frac{1}{\text{Part B}} = \frac{1}{\text{Together}} \]

Used for: Pipes filling tanks, people painting rooms, machines producing parts.

System Decrypt

Unlock the system by solving for \( x \):

\[ \frac{2}{x} = \frac{x - 3}{2} \]

Cross Multiply? \( 4 = x(x-3) \)

Quadratic Form \( x^2 - 3x - 4 = 0 \)

Target Code \( x = 4, -1 \)

System Override Worksheet

SESSION: EQ-ESC-03

System Override Log

Operator: ____________________

Status: LOCKED

Objective: Solving Rational Equations & Detecting Extraneous Viruses

Level 01: Clearing the Line

Multiply by the LCD to clear denominators and solve for the variable.

1.1

\[ \frac{x}{3} + \frac{1}{6} = \frac{5}{6} \]

1.2

\[ \frac{4}{x} + \frac{1}{2} = \frac{10}{x} \]

1.3

\[ \frac{2}{x - 3} = \frac{1}{x + 1} \]

SESSION: EQ-ESC-03

Level 02: Virus Detection (Extraneous)

Solve the equations below. You must check your answers for extraneous solutions (values that make any denominator zero).

2.1

\[ \frac{x}{x - 4} = \frac{4}{x - 4} + 2 \]

2.2

\[ \frac{2}{x + 1} + \frac{x}{x - 1} = 1 \]

Level 03: Workforce Protocol

Mission Specs:

Use the formula: \( \frac{1}{A} + \frac{1}{B} = \frac{1}{\text{Together}} \)

3.1 System A can process a data file in 4 hours. System B can process the same file in 6 hours. How long would it take if both systems worked together?

Check D-Noms

Clearance Granted

System Override Answer Key

KEY: EQ-ESC-03

System Override Answer Key

Solving Rational Equations & Identifying Extraneous Solutions

Level 01: Clearing the Line

1.1 Result:

\[ x = 2 \]

Process: \( 6 \left( \frac{x}{3} + \frac{1}{6} \right) = 6 \left( \frac{5}{6} \right) \implies 2x + 1 = 5 \implies 2x = 4 \]

1.2 Result:

\[ x = 12 \]

Process: \( 2x \left( \frac{4}{x} + \frac{1}{2} \right) = 2x \left( \frac{10}{x} \right) \implies 8 + x = 20 \]

1.3 Result:

\[ x = -5 \]

Process: \( 2(x+1) = 1(x-3) \implies 2x + 2 = x - 3 \implies x = -5 \]

Level 02: Virus Detection

2.1 Solution Analysis:

Warning

\[ \text{No Solution (Extraneous)} \]

Multiplying by \( (x-4) \) gives: \( x = 4 + 2(x-4) \implies x = 4 + 2x - 8 \implies x = 2x - 4 \implies \mathbf{x = 4} \).
However, \( \mathbf{x = 4} \) makes the original denominator \( \mathbf{4 - 4 = 0} \). Therefore, the solution is extraneous.

2.2 Result:

\[ x = \frac{1}{3} \]

LCD is \( (x+1)(x-1) \). Equation becomes: \( 2(x-1) + x(x+1) = 1(x+1)(x-1) \).
\( 2x - 2 + x^2 + x = x^2 - 1 \implies x^2 + 3x - 2 = x^2 - 1 \implies 3x = 1 \).

Level 03: Workforce Protocol

3.1 Solution:

Equation Setup:

\[ \frac{1}{4} + \frac{1}{6} = \frac{1}{x} \]

Solve:

LCD is \( 12x \).
\( 12x \left( \frac{1}{4} \right) + 12x \left( \frac{1}{6} \right) = 12x \left( \frac{1}{x} \right) \)
\( 3x + 2x = 12 \)
\( 5x = 12 \)
\( x = \frac{12}{5} \text{ or } 2.4 \)

Final Result: 2.4 Hours (or 2 hours and 24 minutes)

All Units Operational // Rational Realms Complete