This economics question tests your understanding of economic models and analysis. The step-by-step answer below applies the relevant framework and explains the reasoning.
ScanSolve AI Answer
Step 1: Identify the least common denominator (LCD). The denominators are x-2 and x+1, so LCD = (x-2)(x+1). Step 2: Rewrite each term with the LCD. (3(x+1) + 2(x-2))/((x-2)(x+1)) = 1 Step 3: Simplify the numerator. (3(x+1) + 2(x-2))/((x-2)(x+1)) = (3x + 3 + 2x - 4)/((x-2)(x+1)) = (5x - 1)/((x-2)(x+1)) = 1 Step 4: Subtract 1 from both sides and simplify. (5x - 1)/((x-2)(x+1)) - 1 = 0 (5x - 1 - (x-2)(x+1))/((x-2)(x+1)) = 0 Step 5: Expand the subtracted term. (x-2)(x+1) = x^2 + x - 2x - 2 = x^2 - x - 2 Numerator: 5x - 1 - (x^2 - x - 2) = 5x - 1 - x^2 + x + 2 = -x^2 + 6x + 1 Step 6: Set numerator equal to zero (denominator ≠ 0). -x^2 + 6x + 1 = 0 x^2 - 6x - 1 = 0 Step 7: Solve the quadratic equation using the quadratic formula. x = 6 ± sqrt((-6)^2 - 4(1)(-1))2 · 1 = 6 ± sqrt(36 + 4)2 = 6 ± sqrt(40)2 = 6 ± 2sqrt(10)2 = 3 ± sqrt(10) Step 8: Check for extraneous solutions by testing in the original equation. For x = 3 + sqrt(10): Denominators 3 + sqrt(10) - 2 = 1 + sqrt(10) > 0, 3 + sqrt(10) + 1 = 4 + sqrt(10) > 0. Valid. For x = 3 - sqrt(10): sqrt(10) ≈ 3.16, so 3 - 3.16 ≈ -0.16 < 0, but denominators 3 - sqrt(10) - 2 = 1 - sqrt(10) < 0, 3 - sqrt(10) + 1 = 4 - sqrt(10) > 0. Both defined and equation holds. Valid. Final answer: x = 3 ± sqrt(10) x = 3 ± sqrt(10)