(5х+1)(3х-1)>(4х-1)(х+2);

Предыдущий
вопрос Следующий
вопрос

Let's first expand both sides of the inequality:

(5x + 1)(3x - 1) > (4x - 1)(x + 2)
15x^2 + 5x - 3x - 1 > 4x^2 + 8x - x - 2
15x^2 + 2x - 1 > 4x^2 + 7x - 2

Now, let's simplify the inequality:

15x^2 + 2x - 1 > 4x^2 + 7x - 2
11x^2 - 5x + 1 > 0

Next, we need to solve for x by factoring the quadratic equation:

(11x - 1)(x - 1) > 0

Now, we have two critical points: x = 1/11 and x = 1. We need to test the intervals created by these critical points to find the solution set for x:

When x < 1/11:
Substitute x = 0 into the inequality:
(11(0) - 1)(0 - 1) > 0
(-1)(-1) > 0
1 > 0
True

When 1/11 < x < 1:
Substitute x = 1/6 into the inequality:
(11(1/6) - 1)(1/6 - 1) > 0
(11/6 - 1)(-5/6) > 0
(5/6)(-5/6) > 0
-25/36 > 0
False

When x > 1:
Substitute x = 2 into the inequality:
(11(2) - 1)(2 - 1) > 0
(22 - 1)(1) > 0
21 > 0
True

Therefore, the solution is x < 1/11 or x > 1.