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The partial fraction decomposition of the rational expression x/((x+1)(x-4)) is (1/5)/(x+1) + (4/5)/(x-4).

The area of the region defined by the inequalities y ≥ x, y > -x + 1, and y < 10 is 90.25 square units.

The system of equations is solved by substitution, expanding, simplifying, using the quadratic formula to find x values, and then substituting back into the equations to find the corresponding y values. The solutions are (3, 4) and (-4, -3).

The solution to the system of equations x + 2y = 5 and 3x - y = 1 is x = 1 and y = 2, found using the elimination method.

The solution for x in the equation involving fractions with (x-1) and (x+1) in the denominator is x = √2 and x = -√2.

The solution to the equation 3^(a^2 - 10a + 25)! = 729^4 is a = 3 or a = 7, found by equating exponents after expressing both sides with the same base and solving the resulting quadratic equation.

To find the difference between ba and ab, where a and b are digits, we analyze the units and hundreds digits, leading to the conclusion that the hundreds digit of the difference is 3.

The solution to the equation 4a³ = 2⁵ is a = 2.

The equation was solved by simplifying the square root, isolating the square root term, squaring both sides, moving all terms to one side, factoring out a common factor, and solving for x, resulting in two solutions: x = 0 and x = 16/81.

The solution to the equation 3^x = 1 and 3^y = 81 is x + y = 4.