Rational Equations

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Answer Key & Explanations

Rational Equations · Grade 9 · Worksheet 2

1. (4x - 7)/(x + 9) = 3 = ? Answer: 34 Solution: Start with (4x - 7)/(x + 9) = 3 Multiply both sides by (x + 9): 4x - 7 = 3(x + 9) Distribute the 3 on the right: 4x - 7 = 3x + 27 Subtract 3x from both sides: x - 7 = 27 Add 7 to both sides: x = 34 Check that x = 34 does not make the denominator zero: 34 + 9 = 43 ≠ 0 The solution is x = 34.
   Full step-by-step solution

   Step 1: Start with (4x - 7)/(x + 9) = 3 Step 2: Multiply both sides by (x + 9): 4x - 7 = 3(x + 9) Step 3: Distribute the 3 on the right: 4x - 7 = 3x + 27 Step 4: Subtract 3x from both sides: x - 7 = 27 Step 5: Add 7 to both sides: x = 34 Step 6: Check that x = 34 does not make the denominator zero: 34 + 9 = 43 ≠ 0 The solution is x = 34.

2. A rectangular garden has a length that is 3 meters more than its width. The area of the garden is 70 square meters. If the width is represented by w meters, which equation can be used to find the width? Solve the equation to find the width of the garden. Answer: 7 Solution: Let the width = \( w \) meters. The length is 3 meters more than the width, so length = \( w + 3 \) meters. Area of a rectangle = length × width.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Define the variables** Let the width = \( w \) meters. The length is 3 meters more than the width, so length = \( w + 3 \) meters. --- **Step 2: Write the area equation** Area of a rectangle = length × width. Given area = 70 square meters. So: \[ (w + 3) \times w = 70 \] --- **Step 3: Write the equation in standard form** \[ w(w + 3) = 70 \] \[ w^2 + 3w = 70 \] \[ w^2 + 3w - 70 = 0 \] This is the equation to find the width. --- **Step 4: Solve the quadratic equation** We solve \( w^2 + 3w - 70 = 0 \) by factoring. Look for two numbers whose product is -70 and whose sum is 3. The numbers are 10 and -7 because: \( 10 \times (-7) = -70 \) \( 10 + (-7) = 3 \) So: \[ w^2 + 3w - 70 = (w + 10)(w - 7) = 0 \] --- **Step 5: Find possible values of w** \[ w + 10 = 0 \quad \text{or} \quad w - 7 = 0 \] \[ w = -10 \quad \text{or} \quad w = 7 \] --- **Step 6: Choose the valid solution** Width cannot be negative, so \( w = -10 \) is not valid. Thus, the width is \( w = 7 \) meters. --- **Final answer:** 7

3. (3x + 2)/(x - 1) = 4 = ? Answer: 6 Solution: Start with the equation (3x + 2)/(x - 1) = 4 Multiply both sides by (x - 1) to eliminate the denominator: 3x + 2 = 4(x - 1) Distribute the 4 on the right side: 3x + 2 = 4x - 4 Subtract 3x from both sides: 2 = x - 4 Add 4 to both sides: 6 = x Check that x = 6 doesn't make the denominator zero (6…
   Full step-by-step solution

   Step 1: Start with the equation (3x + 2)/(x - 1) = 4 Step 2: Multiply both sides by (x - 1) to eliminate the denominator: 3x + 2 = 4(x - 1) Step 3: Distribute the 4 on the right side: 3x + 2 = 4x - 4 Step 4: Subtract 3x from both sides: 2 = x - 4 Step 5: Add 4 to both sides: 6 = x Step 6: Check that x = 6 doesn't make the denominator zero (6 - 1 = 5 ≠ 0) The solution is x = 6.

4. Hana is baking cookies for a school bake sale. The recipe calls for 2 cups of flour to make 4 dozen cookies. Hana wants to make 16 dozen cookies. How many cups of flour are needed? Answer: 8 Solution: The ratio is 2 cups per 4 dozen. Step 2: Set up proportion: cups / 16 = 2 / 4. Step 3: Cross-multiply: cups * 4 = 2 * 16.
   Full step-by-step solution

   Step 1: The ratio is 2 cups per 4 dozen. Step 2: Set up proportion: cups / 16 = 2 / 4. Step 3: Cross-multiply: cups * 4 = 2 * 16. Step 4: cups = (2 * 16) / 4. Step 5: cups = 8. The answer is 8 cups.

5. (x + 3)/(x - 2) = 4 = ? Answer: 11/3 Solution: (x + 3)/(x - 2) = 4 Identify the domain restriction. The denominator x - 2 cannot be zero, so x ≠ 2. Eliminate the fraction by multiplying both sides by (x - 2).
   Full step-by-step solution

   We are given the equation: (x + 3)/(x - 2) = 4 Step 1: Identify the domain restriction. The denominator x - 2 cannot be zero, so x ≠ 2. Step 2: Eliminate the fraction by multiplying both sides by (x - 2). (x + 3)/(x - 2) = 4 Multiply both sides by (x - 2): x + 3 = 4(x - 2) Step 3: Expand the right-hand side. x + 3 = 4x - 8 Step 4: Rearrange terms to isolate x. Subtract x from both sides: 3 = 3x - 8 Step 5: Add 8 to both sides. 3 + 8 = 3x 11 = 3x Step 6: Divide both sides by 3. x = 11/3 Step 7: Check the domain. x = 11/3 is not equal to 2, so it is valid. Step 8: Check in the original equation (optional but good practice). Left-hand side: (11/3 + 3)/(11/3 - 2) = (11/3 + 9/3)/(11/3 - 6/3) = (20/3)/(5/3) = (20/3) * (3/5) = 20/5 = 4 Matches the right-hand side. Final answer: x = 11/3

6. Liam is designing a rectangular garden with an area of 48 square meters. He wants the length to be 4 meters more than the width. What are the dimensions of Liam's garden? Answer: width = 6 meters, length = 10 meters Solution: These can be modeled using quadratic equations, where you set up an equation based on the area formula and the given relationship, then solve for the variable representing one dimension.
   Full step-by-step solution

   Many real-world geometry problems involve finding dimensions when given area and a relationship between length and width. These can be modeled using quadratic equations, where you set up an equation based on the area formula and the given relationship, then solve for the variable representing one dimension.

7. A chemical reaction follows the model y = 120/(x+2), where y is the reaction rate in moles per minute and x is the temperature in degrees Celsius. At what temperature will the reaction rate be 15 moles per minute? Answer: 6 Solution: Set up the equation using the given information: 15 = 120/(x+2) Multiply both sides by (x+2) to eliminate the denominator: 15(x+2) = 120 Distribute the 15: 15x + 30 = 120 Subtract 30 from both sides: 15x = 90 Divide both sides by 15: x = 6 The temperature is 6 degrees Celsius.
   Full step-by-step solution

   Step 1: Set up the equation using the given information: 15 = 120/(x+2) Step 2: Multiply both sides by (x+2) to eliminate the denominator: 15(x+2) = 120 Step 3: Distribute the 15: 15x + 30 = 120 Step 4: Subtract 30 from both sides: 15x = 90 Step 5: Divide both sides by 15: x = 6 Step 6: The temperature is 6 degrees Celsius. The answer is 6.

8. Sarah is working on a science project and needs to mix a cleaning solution. The instructions say that for every 4 milliliters of water, you need to add 4 milliliters of vinegar to make the solution. If Sarah has 42 milliliters of water, how much vinegar is needed to maintain the same ratio? Answer: 42 Solution: The ratio of vinegar to water is 4:4. Step 2: Set up the proportion: vinegar / 42 = 4 / 4. Step 3: Cross-multiply: vinegar * 4 = 42 * 4.
   Full step-by-step solution

   Step 1: The ratio of vinegar to water is 4:4. Step 2: Set up the proportion: vinegar / 42 = 4 / 4. Step 3: Cross-multiply: vinegar * 4 = 42 * 4. Step 4: Solve for vinegar: vinegar = (42 * 4) / 4. Step 5: vinegar = 42. The answer is 42 milliliters.

9. (2x + 5)/(x - 3) = 7 = ? Answer: x = 26/5 Solution: (2x + 5)/(x - 3) = 7 Identify the domain restriction. The denominator x - 3 cannot be zero, so x ≠ 3. Eliminate the fraction by multiplying both sides by (x - 3).
   Full step-by-step solution

   We are given the equation: (2x + 5)/(x - 3) = 7 Step 1: Identify the domain restriction. The denominator x - 3 cannot be zero, so x ≠ 3. Step 2: Eliminate the fraction by multiplying both sides by (x - 3). (2x + 5)/(x - 3) * (x - 3) = 7 * (x - 3) This simplifies to: 2x + 5 = 7(x - 3) Step 3: Expand the right-hand side. 2x + 5 = 7x - 21 Step 4: Move all terms involving x to one side and constants to the other. Subtract 2x from both sides: 5 = 5x - 21 Step 5: Add 21 to both sides. 5 + 21 = 5x 26 = 5x Step 6: Solve for x. x = 26/5 Step 7: Check the domain restriction. x = 26/5 is not equal to 3, so it is valid. Final answer: x = 26/5