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Math Help - Rational Inequalities

  1. #1
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    Question Rational Inequalities

    Solve each inequality by graphing, using technology. Express your answer to one decimal place.

    a) 2x+1/x-3 <= x


    Okay, I need help solving this. Am I on the right track?

    1. Make equal to zero : x- 2x+1/x-3 <= 0

    2. "combine terms"? : x(x-3)-2x+1 / x-3 <= 0
    ............................. x^2-5x+1 / x-3 ........up to here is this correct?

    3. Now I look for the "roots" or "zeros" for numerator and denominator?
    ....N: x can't be ____?
    ....D: x can't be +3.


    4. Put this (your roots/zeros) on a number line. Test numbers that are in between each root on the number line?
    ...f(-2) (a#less than 3) = .....
    ...f(4) (a# greater than 3)= .....

    ...Is this the right process? I've been having trouble with inequalities.
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    Re: Rational Inequalities

    Quote Originally Posted by tdotodot View Post
    3. Now I look for the "roots" or "zeros" for numerator and denominator?
    ....N: x can't be ____?
    ....D: x can't be +3.
    Solve the numerator for 0 anyway. The values this gives are also "critical points" for your intervals. (They typically are points where the function goes from + to - or - to +.)

    -Dan
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    Re: Rational Inequalities

    Quote Originally Posted by topsquark View Post
    Solve the numerator for 0 anyway. The values this gives are also "critical points" for your intervals. (They typically are points where the function goes from + to - or - to +.)

    -Dan
    Okay, here's what I did.

    Numerator = 0 ->x=4.8 and x=0.2
    Denominator = 0 -> x=3

    Number line: |-----0.2-------3-------4.8-------|

    Checked each number: 0.2 works , 3 does not work , 4.8 works.

    Tested values in between each interval.
    f(0.1) <=0
    f(1) <=0
    f(4) <=0
    f(5) <=0

    All integers tested between each interval were less than or equal to zero.

    So then my final answer would be: (-infiniti,2) U (4, +infiniti)

    Is that right?
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    Re: Rational Inequalities

    Quote Originally Posted by tdotodot View Post
    Solve each inequality by graphing, using technology. Express your answer to one decimal place.

    a) 2x+1/x-3 <= x


    Okay, I need help solving this. Am I on the right track?

    1. Make equal to zero : x- 2x+1/x-3 <= 0
    This is not quite right. You had \dfrac{2x+1}{x-3} \le x. Subtract \dfrac{2x+1}{x-3} from each side to get 0 \le x - \dfrac{2x+1}{x-3}. Your inequality is flipped the other way.

    Quote Originally Posted by tdotodot View Post
    2. "combine terms"? : x(x-3)-2x+1 / x-3 <= 0
    ............................. x^2-5x+1 / x-3 ........up to here is this correct?
    Not quite. x-\dfrac{2x+1}{x-3} = \dfrac{x(x-3) - (2x+1)}{x-3} = \dfrac{x^2-3x-2x-1}{x-3} = \dfrac{x^2-5x-1}{x-3}. You had x^2-5x+1 in the numerator.
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    Re: Rational Inequalities

    Quote Originally Posted by SlipEternal View Post
    This is not quite right. You had \dfrac{2x+1}{x-3} \le x. Subtract \dfrac{2x+1}{x-3} from each side to get 0 \le x - \dfrac{2x+1}{x-3}. Your inequality is flipped the other way.


    Not quite. x-\dfrac{2x+1}{x-3} = \dfrac{x(x-3) - (2x+1)}{x-3} = \dfrac{x^2-3x-2x-1}{x-3} = \dfrac{x^2-5x-1}{x-3}. You had x^2-5x+1 in the numerator.
    Okay, so here is what I get now...

    Numerator = 0 -> x=5.2 and x=-0.2
    Denominator = 0 ->x=3

    |--------(-0.2)-----------(3)----------(5.2)-----|

    Test Values: (-0.2) = -0.0125 ... is Not greater than or equal to zero. *I am plugging this into "0 <= x^2-5x-1 / x-3"*
    ................... (3) = undefined
    ....................(5.2) = greater than or equal to zero.

    Then pick values between each interval:
    f(-1) -> 1.25 which is >= 0
    f(1) -> 2.5 which is >= 0
    f(4) -> -5 which is < 0
    f(6) ->1.67 which is >= 0


    So back to the original equation now?? ... 2x+1 / x-3 <= 0
    .................................................. ......when is it less than or equal to zero?
    .................................................. ......(4, 5.1)?
    .................................................. ......I'm guessing the answer is wrong? Can someone please explain this to me?
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    Re: Rational Inequalities

    Quote Originally Posted by tdotodot View Post
    Solve each inequality by graphing, using technology. Express your answer to one decimal place.
    a) 2x+1/x-3 <= x.
    Look at this solution.
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