To solve this inequality, we first need to find the critical points where the numerator, denominator, or both are equal to zero.
Setting the numerator equal to zero: x^2 - 1.5x - 7 = 0 This is a quadratic equation that can be factored as: (x - 4)(x + 1.75) = 0 So the critical points from the numerator are x = 4 and x = -1.75.
Setting the denominator equal to zero: (x - 4)^2 = 0 This only has one critical point, x = 4.
Now, we can test the intervals between these critical points to see where the expression is greater than zero.
Interval 1: (-∞, -1.75) Pick a test point within this interval, for example x = -2: Plugging -2 into the expression, we get: ((-2)^2 - 1.5*(-2) - 7)/( -2 - 4)^2 = (4 + 3 - 7)/( (-2) - 4)^2 = (0)/36 = 0 Since the expression is not greater than zero at x = -2, this interval is not part of the solution.
Interval 2: (-1.75, 4) Pick a test point within this interval, for example x = 0: Plugging 0 into the expression, we get: ((0)^2 - 1.5*(0) - 7)/(0 - 4)^2 = (-7)/16 which is less than zero. So this interval is not part of the solution.
Interval 3: (4, ∞) Pick a test point within this interval, for example x = 5: Plugging 5 into the expression, we get: ((5)^2 - 1.5*(5) - 7)/(5 - 4)^2 = (12.5)/1 = 12.5 Since the expression is greater than zero at x = 5, this interval is part of the solution.
Therefore, the solution to the inequality ((x^2-1.5x-7)/(x-4)^2) > 0 is x ∈ (4, ∞).
To solve this inequality, we first need to find the critical points where the numerator, denominator, or both are equal to zero.
Setting the numerator equal to zero:
x^2 - 1.5x - 7 = 0
This is a quadratic equation that can be factored as: (x - 4)(x + 1.75) = 0
So the critical points from the numerator are x = 4 and x = -1.75.
Setting the denominator equal to zero:
(x - 4)^2 = 0
This only has one critical point, x = 4.
Now, we can test the intervals between these critical points to see where the expression is greater than zero.
Interval 1: (-∞, -1.75)
Pick a test point within this interval, for example x = -2:
Plugging -2 into the expression, we get: ((-2)^2 - 1.5*(-2) - 7)/( -2 - 4)^2 = (4 + 3 - 7)/( (-2) - 4)^2 = (0)/36 = 0
Since the expression is not greater than zero at x = -2, this interval is not part of the solution.
Interval 2: (-1.75, 4)
Pick a test point within this interval, for example x = 0:
Plugging 0 into the expression, we get: ((0)^2 - 1.5*(0) - 7)/(0 - 4)^2 = (-7)/16 which is less than zero. So this interval is not part of the solution.
Interval 3: (4, ∞)
Pick a test point within this interval, for example x = 5:
Plugging 5 into the expression, we get: ((5)^2 - 1.5*(5) - 7)/(5 - 4)^2 = (12.5)/1 = 12.5
Since the expression is greater than zero at x = 5, this interval is part of the solution.
Therefore, the solution to the inequality ((x^2-1.5x-7)/(x-4)^2) > 0 is x ∈ (4, ∞).