Lesson 6 Supplementary Exercises 1. Solve for x: (2x - 1)/(x + 3) = 5 2. Solve for x: (x ^ 2)/(x - 3) = (x + 2)/(2x - 5) 3. Solve for x: (x - 1)/(x + 3) > 0 4. Solve for x: 1/x < 1/(x - 3) 5. Solve for x: (x ^ 2 - x - 30)/(x - 1) >= 0​

Let's solve each of these equations: 1. Solve for x: (2x-1)/(x+3) = 5 - Multiply both sides by (x+3): 2x - 1 = 5(x + 3)- Distribute: 2x - 1 = 5x + 15- Subtract 2x from both sides: -1 = 3x + 15- Subtract 15 from both sides: -16 = 3x- Divide both sides by 3: x = -16/3 2. Solve for x: x²/(x-3) = (x+2)/(2x-5) - Cross-multiply: x²(2x-5) = (x+2)(x-3)- Expand: 2x³ - 5x² = x² - x - 6- Move all terms to one side: 2x³ - 6x² + x + 6 = 0- This is a cubic equation, and solving it directly can be complex. We can try factoring:- Notice that x = -1 is a solution (you can substitute to verify).- Factor out (x + 1): (x + 1)(2x² - 8x + 6) = 0- Factor the quadratic: (x + 1)(2)(x - 1)(x - 3) = 0- Solutions: x = -1, x = 1, x = 3

Answered by aikaluv67 | 2024-09-07