(4-x)(5x+1)/(-x-4)(5x-6)<0

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вопрос

To solve this inequality, let's first determine the critical points by setting the numerator and denominator equal to zero:

For (4-x)(5x+1)/(-x-4)(5x-6) < 0:

Numerator: (4-x)(5x+1) = 0
4-x = 0 or 5x+1 = 0
x = 4 or x = -1/5

Denominator: (-x-4)(5x-6) = 0
-x-4 = 0 or 5x-6 = 0
x = -4 or x = 6/5

So the critical points are x = -4, -1/5, 4, and 6/5.

Next, we create intervals on the number line using these critical points to test the sign of the expression (4-x)(5x+1)/(-x-4)(5x-6) in each interval.

-∞ to -4:
Choose x = -5: (4-(-5))(5(-5)+1)/(-(-5)-4)(5(-5)-6) = (9)(-24)/(-1)(-31) > 0

-4 to -1/5:
Choose x = -1: (4-(-1))(5(-1)+1)/(-(-1)-4)(5(-1)-6) = (5)(-4)/3(-11) < 0

-1/5 to 4:
Choose x = 0: (4-0)(5(0)+1)/(-0-4)(5(0)-6) = (4)(1)/(-4)(-6) > 0

4 to 6/5:
Choose x = 5: (4-5)(5(5)+1)/(-5-4)(5(5)-6) = (-1)(26)/(-9)(19) < 0

6/5 to ∞:
Choose x = 7: (4-7)(5(7)+1)/(-7-4)(5(7)-6) = (-3)(36)/(-11)(29) > 0

Therefore, the solution to the inequality (4-x)(5x+1)/(-x-4)(5x-6) < 0 is x ∈ (-4, -1/5) U (6/5, 4).