SOLUTION: Solve the inequality. (Enter your answer using interval notation.) (1)/(x+1)+(1)/(x+8) less than or equal to (1)/(x+9)

Algebra ->  Inequalities -> SOLUTION: Solve the inequality. (Enter your answer using interval notation.) (1)/(x+1)+(1)/(x+8) less than or equal to (1)/(x+9)      Log On


   

Question 1086390: Solve the inequality. (Enter your answer using interval notation.)
(1)/(x+1)+(1)/(x+8) less than or equal to (1)/(x+9)
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
1%2F%28x%2B1%29%2B%281%29%2F%28x%2B8%29%22%22%3C=%22%221%2F%28x%2B9%29

We get 0 on the right:

1%2F%28x%2B1%29%2B%281%29%2F%28x%2B8%29-1%2F%28x%2B9%29%22%22%3C=%22%220

After getting LCD and simplifying, we have:

%28x%5E2%2B18x%2B73%29%2F%28%28x+%2B+1%29%28x+%2B+8%29%28x+%2B+9%29%29%22%22%3C=%22%220

To find all critical values, we set the numerator and denominator
equal to 0 and solve for x:

We set the numerator = 0 and solve for x:

x%5E2%2B18x%2B73=0, which we solve by the quadratic formula 
and get 

-9+%2B-+2+sqrt%282%29 which are critical numbers 
approximately -11.8 and -6.2.

We set the denominator = 0, %28x-1%29%28x%2B8%29%28x%2B9%29=0 and solve for 
x and get -1,-8, and -9.

So we place all the critical numbers on a number line, guessing
about where -11.8 and -6.2 should go between integers.

-----------o----------o---o------o--------------------o--------
-14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1

We pick the most convenient test value in each interval, and
substitute it into %28x%5E2%2B18x%2B73%29%2F%28%28x+%2B+1%29%28x+%2B+8%29%28x+%2B+9%29%29
and if we get a non-positive number, we include that interval
in the solution.  If we get a positive number we do not.

For interval %28matrix%281%2C3%2C-infinity%2C%22%2C%22%2C-9-2sqrt%282%29%29%29
we choose test value -12, substitute it, and get a negative
number, so we include that interval.

For interval %28matrix%281%2C3%2C-9-2sqrt%282%29%2C%22%2C%22%2C-9%29%29
we choose test value -10, substitute it, and get a positive
number, so we do not include that interval.

For interval %28matrix%281%2C3%2C-9%2C%22%2C%22%2C-8%29%29
we choose test value -8.5, substitute it, and get a negative
number, so we include that interval.

For interval %28matrix%281%2C3%2C-8%2C%22%2C%22%2C-9%2B2sqrt%282%29%29%29
we choose test value -7, substitute it, and get a positive
number, so we do not include that interval.

For interval %28matrix%281%2C3%2C-9%2B2sqrt%282%29%2C%22%2C%22%2C-1%29%29
we choose test value -2, substitute it, and get a negative
number, so we include that interval.

For interval %28matrix%281%2C3%2C-1%2C%22%2C%22%2Cinfinity%29%29
we choose test value 0, substitute it, and get a positive
number, so we do not include that interval.

Next, we include the critical values which cause the numerator
to be 0, which are -9+%2B-+2sqrt%282%29.

Thus the solution is 



Edwin