Rational Expressions

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Answer Key & Explanations

Rational Expressions · Grade 9 · Worksheet 2

1. (10/(x+5)) + (15/(x-5)) = ? Answer: (25x+25)/(x²-25) Solution: Identify the denominators: (x+5) and (x-5) The common denominator is (x+5)(x-5) = x²-25 Rewrite each fraction with the common denominator: 10/(x+5) = 10(x-5)/(x²-25) 15/(x-5) = 15(x+5)/(x²-25) Add the numerators: 10(x-5) + 15(x+5) = 10x-50 + 15x+75 = 25x+25 Write the result: (25x+25)/(x²-25) The…
   Full step-by-step solution

   Step 1: Identify the denominators: (x+5) and (x-5) Step 2: The common denominator is (x+5)(x-5) = x²-25 Step 3: Rewrite each fraction with the common denominator: 10/(x+5) = 10(x-5)/(x²-25) 15/(x-5) = 15(x+5)/(x²-25) Step 4: Add the numerators: 10(x-5) + 15(x+5) = 10x-50 + 15x+75 = 25x+25 Step 5: Write the result: (25x+25)/(x²-25) The simplified answer is (25x+25)/(x²-25).

2. Isabella is making a special fruit punch for a party. She needs to combine two different juice mixtures. The first mixture requires (x^2 - 81)/(x^2 + 11x + 24) cups of concentrate, and the second mixture requires (x + 9)/(x + 8) cups of concentrate. What is the total amount of concentrate Isabella needs when she multiplies these two amounts together? Answer: 1 Solution: Write the multiplication problem: [(x^2 - 81)/(x^2 + 11x + 24)] × [(x + 9)/(x + 8)] Factor all polynomials: x^2 - 81 = (x - 9)(x + 9), x^2 + 11x + 24 = (x + 3)(x + 8) Rewrite the expression with factors: [(x - 9)(x + 9)/((x + 3)(x + 8))] × [(x + 9)/(x + 8)] Multiply numerators and denominators:…
   Full step-by-step solution

   Step 1: Write the multiplication problem: [(x^2 - 81)/(x^2 + 11x + 24)] × [(x + 9)/(x + 8)] Step 2: Factor all polynomials: x^2 - 81 = (x - 9)(x + 9), x^2 + 11x + 24 = (x + 3)(x + 8) Step 3: Rewrite the expression with factors: [(x - 9)(x + 9)/((x + 3)(x + 8))] × [(x + 9)/(x + 8)] Step 4: Multiply numerators and denominators: (x - 9)(x + 9)(x + 9)/((x + 3)(x + 8)(x + 8)) Step 5: Cancel common factors: (x + 9) in numerator and denominator cancel Step 6: Simplified expression: (x - 9)(x + 9)/((x + 3)(x + 8)) Step 7: Notice this equals (x^2 - 81)/(x^2 + 11x + 24) which was our original first fraction Step 8: The expression simplifies to 1 when we multiply by the reciprocal Final answer: 1

3. Sophia is making a special fruit punch for a party. The recipe calls for (x^2 - 16)/(x^2 + 6x + 9) cups of cranberry juice and (x + 4)/(x + 3) cups of orange juice. She needs to multiply these two quantities to determine the total amount of juice base required. What is the simplified product of these two rational expressions? Answer: B. (x - 4)/(x + 3)^2 Solution: When multiplying rational expressions, factor all numerators and denominators completely first. Look for special patterns like difference of squares (a^2 - b^2 = (a+b)(a-b)) and perfect square trinomials (a^2 + 2ab + b^2 = (a+b)^2).
   Full step-by-step solution

   When multiplying rational expressions, factor all numerators and denominators completely first. Look for special patterns like difference of squares (a^2 - b^2 = (a+b)(a-b)) and perfect square trinomials (a^2 + 2ab + b^2 = (a+b)^2). After factoring, cancel any common factors that appear in both numerator and denominator. This process simplifies the product to its lowest terms.

4. Emma is making lemonade for a party and needs to mix two different concentrations of lemon juice. She has (x^2 - 9)/(x + 5) cups of strong lemon juice and needs to multiply this by (x + 5)/(x - 3) to get the right amount for her recipe. How many cups of lemon juice will Emma use in her lemonade after performing this multiplication? Answer: x + 3 Solution: Write the multiplication problem: [(x^2 - 9)/(x + 5)] × [(x + 5)/(x - 3)] Factor x^2 - 9 as (x + 3)(x - 3) using the difference of squares formula Rewrite the expression: [(x + 3)(x - 3)/(x + 5)] × [(x + 5)/(x - 3)] Cancel the common factors (x + 5) and (x - 3) The simplified expression is x + 3…
   Full step-by-step solution

   Step 1: Write the multiplication problem: [(x^2 - 9)/(x + 5)] × [(x + 5)/(x - 3)] Step 2: Factor x^2 - 9 as (x + 3)(x - 3) using the difference of squares formula Step 3: Rewrite the expression: [(x + 3)(x - 3)/(x + 5)] × [(x + 5)/(x - 3)] Step 4: Cancel the common factors (x + 5) and (x - 3) Step 5: The simplified expression is x + 3 Emma will use x + 3 cups of lemon juice in her lemonade.

5. (x² - 144)/(x + 12) = ? Answer: x - 12 Solution: Recognize that the numerator x² - 144 is a difference of squares Factor the numerator: x² - 144 = (x + 12)(x - 12) Rewrite the expression: (x + 12)(x - 12)/(x + 12) Cancel the common factor (x + 12) from numerator and denominator The simplified expression is x - 12 The answer is x - 12.
   Full step-by-step solution

   Step 1: Recognize that the numerator x² - 144 is a difference of squares Step 2: Factor the numerator: x² - 144 = (x + 12)(x - 12) Step 3: Rewrite the expression: (x + 12)(x - 12)/(x + 12) Step 4: Cancel the common factor (x + 12) from numerator and denominator Step 5: The simplified expression is x - 12 The answer is x - 12.

6. (x² - 81)/(x + 9) = ? Answer: x - 9 Solution: Factor the numerator: x² - 81 = (x + 9)(x - 9) Rewrite the expression: (x + 9)(x - 9)/(x + 9) Cancel the common factor (x + 9) from numerator and denominator The simplified expression is x - 9 The answer is x - 9.
   Full step-by-step solution

   Step 1: Factor the numerator: x² - 81 = (x + 9)(x - 9) Step 2: Rewrite the expression: (x + 9)(x - 9)/(x + 9) Step 3: Cancel the common factor (x + 9) from numerator and denominator Step 4: The simplified expression is x - 9 The answer is x - 9.

7. Emma is designing a rectangular garden with length (x² - 9)/(x - 3) meters and width (x + 3)/2 meters. What is the area of the garden when x = 7? Answer: 50 Solution: Area = [(x² - 9)/(x - 3)] × [(x + 3)/2] Factor x² - 9 as (x - 3)(x + 3) Area = [(x - 3)(x + 3)/(x - 3)] × [(x + 3)/2] Cancel (x - 3) from numerator and denominator Area = (x + 3) × [(x + 3)/2] = (x + 3)²/2 Substitute x = 7: (7 + 3)²/2 = (10)²/2 = 100/2 = 50 The area is 50 square meters
   Full step-by-step solution

   Step 1: The area of a rectangle is length × width Step 2: Area = [(x² - 9)/(x - 3)] × [(x + 3)/2] Step 3: Factor x² - 9 as (x - 3)(x + 3) Step 4: Area = [(x - 3)(x + 3)/(x - 3)] × [(x + 3)/2] Step 5: Cancel (x - 3) from numerator and denominator Step 6: Area = (x + 3) × [(x + 3)/2] = (x + 3)²/2 Step 7: Substitute x = 7: (7 + 3)²/2 = (10)²/2 = 100/2 = 50 Step 8: The area is 50 square meters