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 zerox^2 - 5x + 6 = (x - 2)(x - 3) = x = 2 or x = 3
Setting the denominator equal to zerox - 4 = 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 < 2Choose x = 1(1^2 - 5*1 + 6)/(1 - 4) = (1 - 5 + 6)/(-3) = 2 > Therefore, the inequality is true for x < 2.
For 2 < x < 3Choose x = 2.5(2.5^2 - 5*2.5 + 6)/(2.5 - 4) = (6.25 - 12.5 + 6)/(-1.5) = -1.5 < Therefore, the inequality is not true for 2 < x < 3.
For 3 < x < 4Choose x = 3.5(3.5^2 - 5*3.5 + 6)/(3.5 - 4) = (12.25 - 17.5 + 6)/(-0.5) = -1.5 < Therefore, the inequality is not true for 3 < x < 4.
For x > 4Choose x = 5(5^2 - 5*5 + 6)/(5 - 4) = (25 - 25 + 6)/1 = 6 > Therefore, the inequality is true for x > 4.
In conclusion, the solution to the inequality isx < 2 or x > 4.
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 =
(x - 2)(x - 3) =
x = 2 or x = 3
Setting the denominator equal to zero
x - 4 =
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 >
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 <
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 <
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 >
Therefore, the inequality is true for x > 4.
In conclusion, the solution to the inequality is
x < 2 or x > 4.