SOLUTION: Could someone please help me solve a rational inequality using the critical point method? 1/x+2>x+2. I know how to use the critical point method, but I am not sure how to get the

Algebra ->  Inequalities -> SOLUTION: Could someone please help me solve a rational inequality using the critical point method? 1/x+2>x+2. I know how to use the critical point method, but I am not sure how to get the       Log On


   

Question 1020598: Could someone please help me solve a rational inequality using the critical point method? 1/x+2>x+2. I know how to use the critical point method, but I am not sure how to get the problem in the correct form first? I appreciate any help you can give me!
Found 2 solutions by josgarithmetic, MathTherapy:
Answer by josgarithmetic(39623) About Me  (Show Source):
You can put this solution on YOUR website!
Make it into relation to zero.

1%2F%28x%2B2%29%3Ex%2B2
1%2F%28x%2B2%29-x-2%3E0
1%2F%28x%2B2%29-x%28x%2B2%29%2F%28x%2B2%29-2%28x%2B2%29%2F%28x%2B2%29%3E0

%281-x%28x%2B2%29-2%28x%2B2%29%29%2F%28x%2B2%29%3E0

%281-x%5E2-2x-2x-4%29%2F%28x%2B2%29%3E0

1-x%5E2-4x-4%29%2F%28x%2B2%29%3E0

%28-x%5E2-4x-3%29%2F%28x%2B2%29%3E0

%28-1%29%28x%5E2%2B4x%2B3%29%2F%28x%2B2%29%3E0

Multiply left and right members by -1 and REVERSE the direction of the inequality symbol...

highlight_green%28%28x%5E2%2B4x%2B3%29%2F%28x%2B2%29%3C0%29-------this is the form which you analyze.

Continue with factorization.
%28%28x%2B1%29%28x%2B3%29%29%2F%28x%2B2%29%3C0
The critical values of x are -1, -2, -3.

Answer by MathTherapy(10555) About Me  (Show Source):
You can put this solution on YOUR website!

Could someone please help me solve a rational inequality using the critical point method? 1/x+2>x+2. I know how to use the critical point method, but I am not sure how to get the problem in the correct form first? I appreciate any help you can give me!
1%2F%28x+%2B+2%29+%3E+x+%2B+2, with x+%3C%3E+-+2

1 = (x + 2)(x + 2) ---- Multiplying each side by LCD, x + 2, and changing INEQUALITY to equality

1+=+x%5E2+%2B+4x+%2B+4

0+=+x%5E2+%2B+4x+%2B+3 --------- Subtracting 1 from each side

x%5E2+%2B+4x+%2B+3+=+0 --------- Reversing sides

(x + 3)(x + 1) = 0

x + 3 = 0         OR       x + 1 = 0

x = - 3           OR       x = - 1 



Thus, the 3 CRITICAL points are: - 2, - 3, and - 1. Arranged in numerical order, this

becomes: - 3, - 2, - 1. From these 3 CRITICAL points, the following 4 intervals have to be tested:

      x < - 3, or x = - 4

- 3 < x < - 2, or x = - 2.5

- 2 < x < - 1, or x = - 1.5

      x > - 1, or x = 0 

When the above intervals are tested, we find that: , are TRUE, and so, these are the solutions. 

The other 2 intervals make the INEQUALITY false, so they're not solutions.