Math Analysis

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

  1. 👍 0
  2. 👎 0
  3. 👁 44
asked by jon
  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)

    1. 👍 0
    2. 👎 0
    posted by Cathy
  2. Note: small correction to my previous answer...

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

    1. 👍 0
    2. 👎 0
    posted by Cathy

Respond to this Question

First Name

Your Response

Similar Questions

  1. Algebra--Correction?

    Solve x^2 -3x is greater than or equal to 10 Would the solutions be x=5 and x= -2? Here's how I figured it. x^2-3x is greater than or equal to 10. I subtracted 10 from both sides. This left me with x^2-3x-10 is greater than or

    asked by Anonymous on October 13, 2009
  2. math

    7. The tennis team is selling key chains as a fundraiser. If its goal is to raise is at least 180, how many key chains must it sell at $2.25 each to meet that goal? Write and solve an inequality. a. 2.25k (greater than or equal

    asked by john on October 18, 2016
  3. math

    please check my work. 1. The number of words (x) Tina can type per minute is at Least 50. write and graph and inequality. (I broke it down to these 2 choices because the A & D was incorrect) B. x is less than or equal to 50 C. x

    asked by kenzie on November 10, 2017
  4. Math

    Hi everyone, my name is Emerson. I was wondering if anybody could help me on a test. I'm stuck on some, but I think I know some others. Thank you! 1. Translate the phrase "nine more than two times a number" in to an algebraic

    asked by Emerson on December 7, 2015
  5. Math

    please check my work. 1. The number of words (x) Tina can type per minute is at Least 50. write and graph and inequality. (I broke it down to these 2 choices because the A & D was incorrect) B. x is less than or equal to 50 C. x

    asked by kenziemack on November 10, 2017
  6. Rational Inequality with Absolute Value

    Here's a question I'm having trouble with: |x|/|x+2|=0 and x+2>0, x must be greater than or equal to zero. When I solved for x in this case, I got x>-4. Reconciling the constraints leaves me back at x>=0. Did I do that correctly?

    asked by Randy on March 26, 2011
  7. math

    9x+5greater than or equal to -40 or 7x+3greater than or equal to -11 solve and graph I got the answer of x greater than or equal -5 or x is greater than or equal to -2 as the solution to this equations, but how do i put this into

    asked by Jen on January 29, 2010
  8. Math

    What is the solution to the nearest tenth to the equation below? -3x -14 less than or equal to -5 * sorry if í wrote the question weird but í couldn't do some equation signs* A. X less than or equal to -3 B. X less than or equal

    asked by Anonymous on August 28, 2017
  9. Math

    Which is equivalent to the inequality shown below? z/2 (less than equal too) - 15 a, x (x greater than equal too) - 7.5 b. x (less than or equal too)-7.5 c. x (greater than equal too) - 30 d. x (less than equal too) - 30 How do I

    asked by G on April 20, 2016
  10. math

    State the null hypothesis, Ho, and the alternative hypothesis, Ha, that would be used to test the following statements. (a) The linear correlation coefficient is positive. Ho: ñ ---Select--- not equal to equal to greater than

    asked by Sue 12 on May 30, 2013
  11. physics

    A beam of electrons with momentum p=10^-23 kg m/s passed through a narrow slit of width d=10^-5 m. What is the uncertainty component of the momentum in the x-direction due to the uncertainty principle (approximately)? a. change in

    asked by Hil on December 23, 2013

More Similar Questions