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Math Help - Inequality

  1. #1
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    Inequality

    I know how to solve the inequality x > (2/(x-1)) but how can I know not to mutiply all by (x-1)^2? That is a method for solving some inequalities.
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    Quote Originally Posted by Stuck Man View Post
    I know how to solve the inequality x > (2/(x-1)) but how can I know not to mutiply all by (x-1)^2? That is a method for solving some inequalities.
    Well, the equation could have been "typed" this way: x(x - 1) > 2 ; get my drift?
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  3. #3
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    Quote Originally Posted by Stuck Man View Post
    I know how to solve the inequality x > (2/(x-1)) but how can I know not to mutiply all by (x-1)^2? That is a method for solving some inequalities.
    you may multiply by (x-1)^2 provided you understand x \ne 1 and you can deal algebraically with the resulting cubic expression.

    In this case, I would prefer to use this method ...

    x > \frac{2}{x-1}

    x - \frac{2}{x-1} > 0

    \frac{x(x-1)}{x-1} -  \frac{2}{x-1} > 0

    \frac{x(x-1) - 2}{x-1} > 0

    \frac{x^2 - x - 2}{x-1} > 0

    \frac{(x-2)(x+1)}{x-1} > 0

    critical values are x = 2 , x = -1 , and x = 0

    these three values are where the expression on the left side of the inequality equals 0 or is undefined.

    last step is to check a value from each interval between the critical values to see if the values in that interval make the original inequality true or false.
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  4. #4
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    I had a mistake with my algebra. So both methods can be used. I don't understand what Wilmer's manipulation achieves.
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  5. #5
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    All I was trying to say is if equation was initially shown as x(x - 1) > 2,
    then you wouldn't even think of: "how can I know not to mutiply all by (x-1)^2?".
    Anyhow, not important...
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    Quote Originally Posted by Stuck Man View Post
    I know how to solve the inequality x > (2/(x-1)) but how can I know not to mutiply all by (x-1)^2? That is a method for solving some inequalities.
    Suppose x<1

    Then if you multiply both sides of the inequality by (x-1)

    you would have to reverse the inequality.

    This is because x-1 is negative when x<1

    5>4 but -5<-4

    The reversal happens if you change the sign of both sides,
    which happens when you multiply both sides by a negative value,
    or divide both sides by a negative value.

    When you are dealing with inequalities, expressions involving x may be positive for certain x
    and negative for other x.

    Hence, if you multiply both sides by a square, you avoid that scenario (since real squares are non-negative).

    Adding and subtracting the same value to both sides does not introduce any sign reversal,
    nor does multiplying by 1, as shown by skeeter.
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