Rational Equations

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Answer Key & Explanations

Rational Equations · Grade 9 · Worksheet 1

1. Liam is analyzing a geometric pattern on a coordinate plane. A rectangle is drawn with vertices at A(0,0), B(10,0), C(10,5), and D(0,5). A diagonal is drawn from A to C. A smaller rectangle is formed by drawing a line from point E on AB to point F on DC, such that the line EF is parallel to AD and passes through the point where the diagonal AC intersects the vertical line x = 4. What is the length of EF? Answer: 5 Solution: The rectangle has vertices A(0,0), B(10,0), C(10,5), D(0,5). The diagonal AC goes from (0,0) to (10,5). Step 2: The line EF is vertical, parallel to AD, so it has constant x.
   Full step-by-step solution

   Step 1: The rectangle has vertices A(0,0), B(10,0), C(10,5), D(0,5). The diagonal AC goes from (0,0) to (10,5). Step 2: The line EF is vertical, parallel to AD, so it has constant x. It passes through the intersection of diagonal AC and the vertical line x=4. Step 3: Find the y-coordinate on diagonal AC at x=4. The diagonal equation is y = (5/10)x = (1/2)x. At x=4, y = (1/2)*4 = 2. Step 4: So the intersection point is (4,2). The vertical line EF goes from this point to the top of the rectangle (same x, y=5) or bottom (y=0). But EF spans from AB (y=0) to DC (y=5) at x=4. Step 5: The length of EF is the vertical distance from y=0 to y=5 at x=4, which is 5 - 0 = 5. The answer is 5.

2. (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.

3. Liam is designing a rectangular garden with a length that is 3 meters more than twice its width. He needs to install a decorative border along the perimeter, which requires 36 meters of border material. What are the dimensions of Liam's garden? Answer: width = 5 m, length = 13 m Solution: Let the width of the garden be \( w \) meters. The length is 3 meters more than twice the width, so: length \( l = 2w + 3 \). \( P = 2 \times (\text{length} + \text{width}) \).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define variables** Let the width of the garden be \( w \) meters. The length is 3 meters more than twice the width, so: length \( l = 2w + 3 \). --- **Step 2: Write the perimeter equation** The perimeter \( P \) of a rectangle is: \( P = 2 \times (\text{length} + \text{width}) \). We are told the perimeter is 36 m: \( 2 \times (l + w) = 36 \). --- **Step 3: Substitute the expression for \( l \)** Substitute \( l = 2w + 3 \) into the perimeter equation: \( 2 \times ( (2w + 3) + w ) = 36 \). --- **Step 4: Simplify inside the parentheses** \( (2w + 3) + w = 3w + 3 \). So: \( 2 \times (3w + 3) = 36 \). --- **Step 5: Divide both sides by 2** \( 3w + 3 = 18 \). --- **Step 6: Subtract 3 from both sides** \( 3w = 15 \). --- **Step 7: Divide by 3** \( w = 5 \). --- **Step 8: Find length** \( l = 2w + 3 = 2 \times 5 + 3 = 10 + 3 = 13 \). --- **Final answer:** Width = 5 m, Length = 13 m.

4. Liam is designing a rectangular garden with an area of 48 square meters. He wants the length to be 2 meters more than the width. To determine the optimal dimensions, he sets up the equation x(x + 2) = 48, where x represents the width in meters. What are the actual dimensions of Liam's garden? Answer: 6 meters by 8 meters Solution: Write down the equation from the problem. Area = 48 m² Length = width + 2 Let width = x Then length = x + 2 Area = width × length x(x + 2) = 48 Expand and rearrange into a standard quadratic equation.
   Full step-by-step solution

   Let's solve this step by step. --- **Step 1: Write down the equation from the problem.** The problem says: Area = 48 m² Length = width + 2 Let width = x Then length = x + 2 Area = width × length x(x + 2) = 48 --- **Step 2: Expand and rearrange into a standard quadratic equation.** x(x + 2) = 48 x² + 2x = 48 x² + 2x − 48 = 0 --- **Step 3: Solve the quadratic equation.** We can factor it: We need two numbers that multiply to −48 and add to +2. Those numbers are +8 and −6. So: (x + 8)(x − 6) = 0 --- **Step 4: Find possible values for x.** x + 8 = 0 → x = −8 x − 6 = 0 → x = 6 Since width cannot be negative, we discard x = −8. So x = 6. --- **Step 5: Find the length.** Length = x + 2 = 6 + 2 = 8 --- **Step 6: Check the area.** Area = 6 × 8 = 48 m² ✔ --- **Final answer:** The dimensions are 6 meters by 8 meters.

5. Mason is analyzing a geometric pattern on a coordinate grid. A right triangle is drawn with vertices at A(0,0), B(12,0), and C(0,9). A rectangle is inscribed inside the triangle such that one side lies along the x-axis from (0,0) to (x,0) and the opposite side is parallel to the x-axis, touching the hypotenuse at a point. The height of the rectangle is given by the expression (9 - (3/4)x). If the area of the rectangle is expressed as A = x(9 - (3/4)x), and the maximum area occurs when the derivative is zero, find the value of x that maximizes the area. Answer: 6 Solution: Identify the line of the hypotenuse. The triangle has vertices at (0,0), (12,0), and (0,9). The hypotenuse connects (12,0) to (0,9).
   Full step-by-step solution

   Step 1: Identify the line of the hypotenuse. The triangle has vertices at (0,0), (12,0), and (0,9). The hypotenuse connects (12,0) to (0,9). Its slope is (9 - 0)/(0 - 12) = 9/(-12) = -3/4. So the equation of the hypotenuse is y = (-3/4)x + 9. Step 2: The height of the rectangle at any x is the y-coordinate on the hypotenuse, which is y = (-3/4)x + 9. This matches the given expression 9 - (3/4)x. Step 3: The area of the rectangle is width times height: A = x * (9 - (3/4)x) = 9x - (3/4)x^2. Step 4: To find the x that maximizes the area, take the derivative of A with respect to x: dA/dx = 9 - (3/2)x. Step 5: Set the derivative equal to zero: 9 - (3/2)x = 0. Solve for x: (3/2)x = 9, so x = 9 * (2/3) = 6. Step 6: Verify this gives a maximum (second derivative is negative). The value of x that maximizes the area is 6. The answer is 6.

6. (2x + 3)/(x - 1) = 5 = ? Answer: 8/3 Solution: We are given the equation: (2x + 3)/(x - 1) = 5. Since the equation is (2x + 3)/(x - 1) = 5, we can multiply both sides by (x - 1) to eliminate the denominator.
   Full step-by-step solution

   We are given the equation: (2x + 3)/(x - 1) = 5. Step 1: Since the equation is (2x + 3)/(x - 1) = 5, we can multiply both sides by (x - 1) to eliminate the denominator. But note: x cannot be 1 because that would make the denominator zero. So, assuming x ≠ 1, multiply both sides by (x - 1): (2x + 3) = 5 * (x - 1) Step 2: Expand the right-hand side: 2x + 3 = 5x - 5 Step 3: Bring like terms together. Subtract 2x from both sides: 3 = 5x - 5 - 2x 3 = 3x - 5 Step 4: Add 5 to both sides: 3 + 5 = 3x 8 = 3x Step 5: Divide both sides by 3: x = 8/3 Step 6: Check if x ≠ 1 (yes, 8/3 is not 1) and substitute back to verify: (2*(8/3) + 3) / (8/3 - 1) = (16/3 + 9/3) / (8/3 - 3/3) = (25/3) / (5/3) = (25/3) * (3/5) = 25/5 = 5. It works. Final answer: x = 8/3

7. (8x - 9)/(x + 7) = 7 = ? Answer: 58 Solution: Multiply both sides by (x + 7): (8x - 9) = 7(x + 7) Distribute the 7: 8x - 9 = 7x + 49 Subtract 7x from both sides: x - 9 = 49 Add 9 to both sides: x = 58 Check: (8*58 - 9)/(58 + 7) = (464 - 9)/65 = 455/65 = 7.
   Full step-by-step solution

   Step 1: Multiply both sides by (x + 7): (8x - 9) = 7(x + 7) Step 2: Distribute the 7: 8x - 9 = 7x + 49 Step 3: Subtract 7x from both sides: x - 9 = 49 Step 4: Add 9 to both sides: x = 58 Step 5: Check: (8*58 - 9)/(58 + 7) = (464 - 9)/65 = 455/65 = 7. The solution is x = 58.