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Solving Rational Inequalities

  1. Mar 5, 2009 #1
    I wanted help on solving Solving Rational Inequalities. I watched this video and i was wondering if that was the best or easiest way. Our teacher taught us a much harder and weirder way using multiple number lines instead of one. If you do watch the video the man shows examples with open dots, what if the dots are closed? I have an idea but im not sure
     
    Last edited by a moderator: Sep 25, 2014
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  3. Mar 6, 2009 #2

    Gib Z

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    Homework Helper

    Personally I don't like the method he uses. Generally, I like to transform rational inequalities into polynomial inequalities. I do this by multiplying by multiplying both sides by some positive value (which preserves the direction of the inequality sign) that will get rid of the denominator, and solve the resulting polynomial inequality.

    Ill address some of the examples of the video to make this clear; his first example is :

    [tex]\frac{3x}{x-4} \leq 8[/tex]. To get rid of the denominator, I multiply by (x-4)^2 (make not here that we already know x=4 can not be a solution) to give [tex] 3x(x-4) \leq 8(x-4)^2[/tex]. Taking all terms to one side, [tex]8(x-4)^2 - 3x(x-4) \geq 0[/tex]. And now by factoring, [tex](x-4)(5x-32) \geq 0[/tex]. Now this last inequality is easy. We could do it by dividing the real line into 3 sections and testing points as done in the video, but I prefer to quickly imagine the graph of the parabola to see when it is greater or equal to zero. Or draw it out if you are uncertain. Doing this gives [tex]x \geq 4, x\leq 32/5[/tex], but since x = 4 can't be a solution, its just Doing this gives [tex]x > 4, x\leq 32/5[/tex].

    The next one is [tex] \frac{2x}{x-5} < \frac{7x}{x-4}[/tex]. To get rid of the denominators, and to preserve the sign, multiply by (x-4)^2(x-5)^2. Don't expand anything, its all easier if its factored. Take everything to one side and pull out common factors to get [tex]x(x-5)(x-4) \left( 2(x-4) - 7(x-5) \right) < 0[/tex]. Now simplify that last term, and we have the factored form of a polynomial. This makes it very easy to sketch as you know the x intercepts. Using this sketch we can see the values of x which satisfy the inequality. Just remember at the end to exclude x=4 and x=5 form the solution.

    The last one: [tex] \frac{5}{x} - \frac{6}{x^2} \geq 0[/tex]. This ones easy, multipling by x^2 gets rid of denominators and preserves the sign, so that it becomes [tex]5x-6 \geq 0[/tex]. Remember to exclude x=0 from the solution set.

    Well. Thats the method I like best.
     
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