(x^2 - 5*x + 6)/(x - 4) > 0

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вопрос

To solve the inequality (x^2 - 5x + 6)/(x - 4) > 0, we first need to find the critical points where the expression is equal to zero or undefined.

Setting the numerator equal to zero:
x^2 - 5x + 6 = 0
(x - 2)(x - 3) = 0
x = 2 or x = 3

Setting the denominator equal to zero:
x - 4 = 0
x = 4

So, the critical points are x = 2, x = 3, and x = 4.

Next, we will test the intervals between these critical points to determine where the inequality holds true.

For x < 2:
Choose x = 1:
(1^2 - 5*1 + 6)/(1 - 4) = (1 - 5 + 6)/(-3) = 2 > 0
Therefore, the inequality is true for x < 2.

For 2 < x < 3:
Choose x = 2.5:
(2.5^2 - 5*2.5 + 6)/(2.5 - 4) = (6.25 - 12.5 + 6)/(-1.5) = -1.5 < 0
Therefore, the inequality is not true for 2 < x < 3.

For 3 < x < 4:
Choose x = 3.5:
(3.5^2 - 5*3.5 + 6)/(3.5 - 4) = (12.25 - 17.5 + 6)/(-0.5) = -1.5 < 0
Therefore, the inequality is not true for 3 < x < 4.

For x > 4:
Choose x = 5:
(5^2 - 5*5 + 6)/(5 - 4) = (25 - 25 + 6)/1 = 6 > 0
Therefore, the inequality is true for x > 4.

In conclusion, the solution to the inequality is:
x < 2 or x > 4.