Algebraic Reasoning Mastery

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Algebraic expressions and equations illustration

📚 Part 1: Simplifying and Factoring Expressions

1. Simplify the expression: 3x + 5x - 2x + 7

2. Factor the expression using the distributive property: 4x + 12

3. Which expression is equivalent to 2(3x + 4)?

6x + 4

6x + 8

5x + 6

2x + 8

4. Simplify: 7y - 3y + 2 + 5y - 8

5. Factor completely: 15a - 10

🌍 Part 2: Real-World Applications

6. A rectangle has a length of (2x + 6) feet and a width of 3 feet. Write an expression for the area and then factor it.

Area expression: _________________

Factored form: _________________

7. Maria buys x notebooks for $3 each and x pens for $2 each. She also buys a backpack for $15. Write an expression for the total cost and simplify it.

Expression: _________________

Simplified: _________________

8. The perimeter of a triangle is represented by the expression 4x + 6x + 2x - 3. Simplify this expression.

9. A parking lot charges $5 per hour plus a $3 entrance fee. If someone parks for h hours, which expression represents the total cost?

5h + 3

5 + 3h

5h - 3

🧠 Part 3: Critical Thinking

10. Explain in your own words why 3(x + 4) and 3x + 12 are equivalent expressions. Use an example with specific numbers to support your explanation.

11. Create your own real-world problem that can be represented by the expression 2x + 8, then factor the expression and explain what each part represents in your problem.

12. Challenge: If the expression 6x + 9y - 3 represents the cost of a school fundraiser, factor out the greatest common factor and explain what this might represent in the real-world context.

Answer Key:

1) 6x + 7; 2) 4(x + 3); 3) 6x + 8; 4) 9y - 6; 5) 5(3a - 2); 6) Area: 3(2x + 6) = 6x + 18, Factored: 6(x + 3); 7) 3x + 2x + 15 = 5x + 15; 8) 12x - 3; 9) 5h + 3; 10-12) Answers will vary