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.
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Here are the solutions to the mathematics questions. The binary operation is defined on the set of real numbers, R, by x y = xy + x + y. a) Find the inverse of x. Step 1: Find the identity element e. The identity element e is such that for any x R, x e = x. x e = xe + x + e Setting this equal to x: xe + x + e = x Subtract x from both sides: xe + e = 0 Factor out e: e(x+1) = 0 For this equation to hold for all x R (where x ≠ -1), the identity element must be e=0. We can verify this: x 0 = x(0) + x + 0 = 0 + x + 0 = x. Also, 0 x = 0(x) + 0 + x = 0 + 0 + x = x. So, the identity element is e=0. Step 2: Find the inverse of x, denoted as x^-1. The inverse x^-1 is such that x x^-1 = e. Since e=0: x x^-1 = 0 Using the definition of the operation: x(x^-1) + x + x^-1 = 0 Factor out x^-1 from the terms containing it: x^-1(x+1) + x = 0 Subtract x from both sides: x^-1(x+1) = -x Divide by (x+1) to solve for x^-1 (provided x+1 ≠ 0, i.e., x ≠ -1): x^-1 = (-x)/(x+1) The inverse of x is (-x)/(x+1) for x ≠ -1. b) Find the solution of x 3 = 19. Step 1: Use the definition of the binary operation. The operation is x y = xy + x + y. Substitute y=3 into the definition: x 3 = x(3) + x + 3 x 3 = 3x + x + 3 x 3 = 4x + 3 Step 2: Set the expression equal to 19 and solve for x. We are given x 3 = 19: 4x + 3 = 19 Subtract 3 from both sides: 4x = 19 - 3 4x = 16 Divide by 4: x = (16)/(4) x = 4 The solution is x=4. Got more? Send 'em!