This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Rewrite the terms with negative exponents as fractions. Recall that x^-1 = (1)/(x). Substitute this into the given expression: 4x^-1 - 5 + xx^-1 - 1 = (4)/(x) - 5 + x(1)/(x) - 1 Step 2: Find a common denominator for the terms in the numerator and the denominator separately. For the numerator: (4)/(x) - 5 + x = (4)/(x) - (5x)/(x) + (x^2)/(x) = (4 - 5x + x^2)/(x) For the denominator: (1)/(x) - 1 = (1)/(x) - (x)/(x) = (1 - x)/(x) Step 3: Substitute these simplified expressions back into the fraction. (x^2 - 5x + 4)/(x)(1 - x)/(x) Step 4: Multiply the numerator by the reciprocal of the denominator. (x^2 - 5x + 4)/(x) × (x)/(1 - x) Step 5: Cancel out the common term x (assuming x ≠ 0). (x^2 - 5x + 4)/(1 - x) Step 6: Factor the quadratic expression in the numerator. We need two numbers that multiply to 4 and add to -5. These numbers are -1 and -4. So, x^2 - 5x + 4 = (x - 1)(x - 4). Step 7: Substitute the factored numerator back into the expression. ((x - 1)(x - 4))/(1 - x) Notice that 1 - x is the negative of x - 1, i.e., 1 - x = -(x - 1). Step 8: Substitute -(x - 1) for 1 - x and simplify. ((x - 1)(x - 4))/(-(x - 1)) Cancel the common term (x - 1) (assuming x - 1 ≠ 0): (x - 4)/(-1) = -(x - 4) = -x + 4 The simplified expression is: 4 - x 3 done, 2 left today. You're making progress.