Question

X-4/x-5 - x-6/x-10 меньше либо равно 2

Answer 1

$\\\frac{x-4}{x-5} - \frac{x-6}{x-10} \leq 2\\ \\\frac{(x-4)(x-10)-(x-6)(x-5)}{(x-5)(x-10)}\leq2|(x-5)(x-10)\\ \\(x-4)(x-10)-(x-6)(x-5)} \leq 2*(x-5)(x-10)\\ \\(x-4)(x-10)-(x-6)(x-5)} - 2*(x-5)(x-10) \leq 0 \\ \\x^2 - 10x -4x + 40 - (x^2 - 5x - 6x + 30) - 2(x^2 - 10x - 5x + 50) \leq0\\ \\x^2 - 14x + 40 - x^2 + 11x - 30 -2x^2 + 30x - 100 \leq2\\ \\-2x^2 - 3x + 30x - 90 \leq0 | * (-1) \\ \\2x^2-27x + 90 \geq 0$
$\\2x^2 - 27x + 90 = 0 \\ \\D = (-27)^2 - 4 * 2 * 90 = 729 - 720 = 9 \\ \\\sqrt{9} = 3 \\ \\x_1 = \frac{27+3}{4} = \frac{30}{4} = 7.5 \\ \\x_2 = \frac{27 - 3}{4} = \frac{24}{4} = 6$
x∈(-∞;6) U (7.5 ; +∞)