SOLUTION: the function (x-1)/((x-4)*(sqrt(x+2))) is negative for x in? A)(1,4) B)(- infinity,4) C)(4, infinity) D)(- infinity,1)

Algebra ->  Rational-functions -> SOLUTION: the function (x-1)/((x-4)*(sqrt(x+2))) is negative for x in? A)(1,4) B)(- infinity,4) C)(4, infinity) D)(- infinity,1)      Log On


   

Question 887921: the function (x-1)/((x-4)*(sqrt(x+2))) is negative for x in?
A)(1,4)
B)(- infinity,4)
C)(4, infinity)
D)(- infinity,1)
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
%28x-1%29%2F%28%28x-4%29%2A%28sqrt%28x%2B2%29%29%29

Identify the critical points and check the sign in each interval. Your critical points are at x from the values of {-2, 1, 4}. Four intervals to check.

MORE EXPLANATION

Look for the critical points. These are the values for x which make the function intersect the x-axis or which make the function undefined. Note that the numerator is 0 if x=1, and therefore this is a root or zero or x-intercept for the function. The value of the function is 0 when x=1. Note that the denominator MUST NOT be zero; so x%3C%3E4 and x%3C%3E-2.
The critical points for x are -2, 1, and 4. The critical points cut the x-axis into four intervals. Also, if you must keep within REAL numbers, you must have any x%3E-2. So really you have domain for x%3E-2 and the interval to the left must not be used for the function.
Test the sign of the function for these intervals on x:
-2%3Cx%3C1;
1%3Cx%3C4;
4%3Cx.
I used order-relation notation, and not "interval" notation. You should understand that the critical points are where the signs of the algebraic factors change. Your question was to find the signs in the different intervals for the function. "Where is the function negative?"