how do you solve the inequality (x+1)/(2-x)>2

Question

as of right now i have the equation set equal to zero. but i am stuck and i am not sure how to continue on.

 

this is what i have:

((x+1)/(2-x))-2=0

Answer 1

Solve (x+1)/(2-x)>2

1.  (2-x)•(x+1)/(2-x) >2 (2-x)                  multiply both sides by the denominator...

_{remember you can always get rid of a denominator by multiply EVERYTHING by that denominator :)}

2. (x+1) > 2(2-x)                                    simplify the right side using the Distributive Property

3. (x+1) > 4 - 2x                                     To solve this inequality, x's on one side, and constants (terms with numbers and no variables).  To do this, add 2x to both sides and subtract one from both sides, leaving you with: 3x > 3

4. 3x > 3, so x>1.  Now we must consider any type of inconsistency of siginifiacnt x values, for instance the one that makes the original denominator zero.

What value of x makes (x+1)/(2-x) undefined (zero on bottom).  Well this is easy.... 2-x = 0, so x=2.  This means that the expression is undefined at x=2, so even though we got x>1 before, we now know x ≠ 2.  Can it equal any number larger than two?  Try 3 and 4 and see if you can find a pattern.

(x+1)/(2-x)>2 for x=3 is      (3+1)/(2-3)>2

                                               4 / -1 > 2  NO GOOD!

(x+1)/(2-x)>2 for x=4 is      (4+1)/(2-4)>2

                                               5 / -2 > 2 NO GOOD!

As you should notice, with any number larger than 2, the left side is negative which will NEVER be >2.

Summary: we found x > 1, that x≠2 or any number larger than two, so our solution is 1 < x < 2.

 

Thanks for your time. Hope this helps :)

 so the answer is 1 < x < 2.

 

Answer 2

You have the beginning of the first part of the problem completed.  You need to solve this for x to get one of your open points on the number line.

(x+1)/(2-x) = 2

x + 1 = 2(2-x)

x + 1 = 4 - 2x

3x = 3

x = 3

The next part of the problem is finding all points at which the denominator is 0.  This will provide the other open circle on the number line.

2 - x = 0

x = 2

If you mark these two points on  a number line, you will have three segments of number line.  x <1, 1<x<2, and x>2.  You need to test a number in each of these domains.  If the answer is correct, then that segment is part of the solution.  If the answer is not correct, then that segment of the number line is not part of the solution.

We will use the points 0, 1.5, and 3.

(0 + 1)/(2 - 1) > 2                             (1.5 + 1)/(2 - 1.5) > 2                         (3 + 1)/(2 -3) > 2

1/1 > 2                                             (2.5)/(.5) > 2                                      (4)/(-1) > 2

1 > 2                                                  5 > 2                                                 -4 > 2

Incorrect (x<1 is not in the solution)  Correct (1<x<2 is in the solution) Incorrect (x >2 is not in the solution

The final answer is 1 < x < 2.