How would I solve: x-1/x+2 is greater than or equal to x/x-2

Answer 1

Subtract the right side from both sides, leaving zero on the right side.

[(x-1)/(x+2)] - [x/(x-2)] >= 0

Get common denominator.
Multiply first term by (x-2)/(x-2).
Multiply second term by (x+2)/(x+2).

[(x-2)(x-1)]/[(x-2)(x+2)] - [(x+2)x]/[(x+2)(x-2)] >= 0

Multiply.
(x^2 - x - 2x + 2)/(x^2 - 4) - (x^2 + 2x)/(x^2 - 4) >= 0

Combine the terms into one rational expression:
(x^2 - 3x + 2 - x^2 - 2x)/(x^2 - 4) >= 0

Simplify:
(-5x + 2)/(x^2 - 4) >= 0

Find the value of x that will satisfy the equality part of the inequality.
Determine what value of x will make this true:
(-5x + 2)/(x^2 - 4) = 0

To do this, determine what value of x will make numerator equal to zero:
-5x + 2 = 0
-5x + 2 - 2 = 0 - 2
-5x = -2
x = 2/5

Thus, when x=2/5, the rational expression on the right side will be equal to zero.
x = 2/5 is a solution

Determine which value(s) of x will make the rational expression undefined.

To do this, determine what value of x will make denominator equal to zero:
x^2 - 4 = 0
x^2 = 4
x = 2, x = -2

Thus, when x = 2 or x = -2, the rational expression on the right side will be undefined.
x = 2 and x = -2 are not solutions

So far, we know that
x = -2 is not a solution
x = 2/5 is a solution
x = 2 is not a solution.

We need to choose four values for x to represent all the possible values of x to determine all the valid solutions.
Choose an x value less than -2
Choose an x value between -2 and 2/5
Choose an x value between 2/5 and 2
Choose an x value greater than 2

Choose an x value less than -2: x = -3
Substitute this into the inequality:
(-5x + 2)/(x^2 - 4) >= 0
(-5(-3) + 2)/((-3)^2 - 4) >= 0
(15 + 2)/(9 - 4) >= 0
17/5 >= 0 is true
So x values less than -2 are solutions.

Choose an x value between -2 and 2/5: x = -1
Substitute this into the inequality:
(-5x + 2)/(x^2 - 4) >= 0
(-5(-1) + 2)/((-1)^2 - 4) >= 0
(5 + 2)/(1 -4) >= 0
7/-3 >= 0 or -7/3 >= 0 is not true
So x values beteen -2 and 2/5 are not solutions.

Choose an x value between 2/5 and 2: x = 1
Substitute this into the inequality:
(-5x + 2)/(x^2 - 4) >= 0
(-5(1) + 2)/((1)^2 - 4) >= 0
(-5 + 2)/(1 - 4) >= 0
-3/-3 >= 0 or 1 >= 0 is true
So x values between 2/5 and 2 are solutions.

Choose an x value greater than 2: x = 3
Substitute this into the inequality:
(-5x + 2)/(x^2 - 4) >= 0
(-5(3) + 2)/((3)^2 - 4) >= 0
(-15 + 2)/(9 - 4) >= 0
-13/5 >= 0 is not true.
So x values greater than 2 are not solutions.

So we now know that:
x values less than -2 are solutions
x = -2 is not a solution
x values beteen -2 and 2/5 are not solutions
x = 2/5 is a solution
x values between 2/5 and 2 are solutions
x = 2 is not a solution
x values greater than 2 are not solutions

Therefore the solutions for this inequality are represented as:

-infinity < x < -2 and 2/5 <= x < 2

(-infinity, -2) and [2/5, 2)

Answer 2

Note: small correction to my previous answer...

Both occurrences of "rational expression on the right side" should be replaced with just "rational expression".