if the inequality is: (x+5/x-2)>0 . "x-5 over x-2 greater or equal to 0". What would be the solution?
a.(-infinity sign, -5] U (2, infinity sign)
b.(-5,2)
c.(-infinity sign, -5) U [2, infinity sign)
d.[-5,2]
My answer was C. Am I correct?
You are looking for those values of x, so that
y = (x+5)/(x-2) lies above the x-axis
first of all, x ≠ 2 or else we are dividing by zero
critical values are -5 and 2
investigate x < -5, say x = -10
then y = -/- = + , that's good
investigate between -5 and 2, say x = 0
y = +/2 < 0 , no good
investigate x > 2 , say x = 5
y = +/+ > 0 , that's good
in my notation:
x< -5 OR x > 2
I will let you sort it out with the new-fangled notation.
To find the solution to the inequality (x+5)/(x-2) > 0, you can follow these steps:
1. Find the potential values of x that make the numerator (x+5) equal to zero. In this case, x = -5.
2. Find the potential values of x that make the denominator (x-2) equal to zero. In this case, x = 2.
3. Plot these values on a number line, dividing it into three intervals: (-infinity, -5), (-5, 2), and (2, infinity).
4. Choose a value from each interval to test the inequality. For example, you can choose -6 from (-infinity, -5), 0 from (-5, 2), and 3 from (2, infinity).
5. Substitute these chosen values into the original inequality and determine whether it satisfies the inequality.
- For x = -6: ((-6+5)/(-6-2)) = -(-1)/(8) = 1/8 > 0 (True)
- For x = 0: ((0+5)/(0-2)) = 5/(-2) = -5/2 < 0 (False)
- For x = 3: ((3+5)/(3-2)) = 8/1 = 8 > 0 (True)
6. Analyze the signs of the tested values. In this case, the values that satisfy the inequality occur when the expression is positive, which is in the interval (-infinity, -5) U (2, infinity).
Therefore, the correct answer is option c. (-infinity, -5) U [2, infinity).