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
Here are the solutions to the problems: 49. Solve |5x+4| 1. Step 1: Rewrite the absolute value inequality. -1 5x+4 1 Step 2: Subtract 4 from all parts of the inequality. -1 - 4 5x 1 - 4 -5 5x -3 Step 3: Divide all parts by 5. (-5)/(5) x (-3)/(5) -1 x -(3)/(5) Step 4: Express the solution in interval notation. x [-1, -(3)/(5)] The correct option is B. 50. Solve |5x-7| > 2. Step 1: Rewrite the absolute value inequality as two separate inequalities. 5x-7 > 2 or 5x-7 < -2 Step 2: Solve the first inequality. 5x-7 > 2 5x > 2+7 5x > 9 x > (9)/(5) Step 3: Solve the second inequality. 5x-7 < -2 5x < -2+7 5x < 5 x < (5)/(5) x < 1 Step 4: Combine the solutions and express in interval notation. x (-, 1) ((9)/(5), ) The correct option is B. 51. Resolve (6x-9)/((x-3)(x-2)) into partial fractions. Step 1: Set up the partial fraction decomposition. (6x-9)/((x-3)(x-2)) = (A)/(x-3) + (B)/(x-2) Step 2: Multiply both sides by (x-3)(x-2). 6x-9 = A(x-2) + B(x-3) Step 3: Solve for A and B. Let x=2: 6(2)-9 = A(2-2) + B(2-3) 12-9 = 0 + B(-1) 3 = -B B = -3 Let x=3: 6(3)-9 = A(3-2) + B(3-3) 18-9 = A(1) + 0 9 = A Step 4: Substitute A and B back into the decomposition. (6x-9)/((x-3)(x-2)) = (9)/(x-3) + (-3)/(x-2) (6x-9)/((x-3)(x-2)) = (9)/(x-3) - (3)/(x-2) The correct option is B. 52. Resolve (8x+4)/(x^2-4) into partial fractions. Step 1: Factor the denominator. x^2-4 = (x-2)(x+2) Step 2: Set up the partial fraction decomposition. (8x+4)/((x-2)(x+2)) = (A)/(x-2) + (B)/(x+2) Step 3: Multiply both sides by (x-2)(x+2). 8x+4 = A(x+2) + B(x-2) Step 4: Solve for A and B. Let x=2: 8(2)+4 = A(2+2) + B(2-2) 16+4 = A(4) + 0 20 = 4A A = 5 Let x=-2: 8(-2)+4 = A(-2+2) + B(-2-2) -16+4 = 0 + B(-4) -12 = -4B B = 3 Step 5: Substitute A and B back into the decomposition. (8x+4)/(x^2-4) = (5)/(x-2) + (3)/(x+2) The correct option is B. 53. Find the general term T_r+1 of the binomial expansion of (2x + (1)/(x))^8. Step 1: Use the binomial theorem formula for the (r+1)-th term: T_r+1 = nr a^n-r b^r. Here, n=8, a=2x, and b=(1)/(x). Step 2: Substitute the values into the formula. T_r+1 = 8r (2x)^8-r ((1)/(x))^r Step 3: Simplify the expression. T_r+1 = 8r 2^8-r x^8-r x^-r T_r+1 = 8r 2^8-r x^8-r-r T_r+1 = 8r 2^8-r x^8-2r The correct option is B.