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Math Help - inequalities of rational expressions

  1. #1
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    inequalities of rational expressions

    referring to the photo attached, can anyone expalin and give suitable numerical example please? i couldnt understand the notes. inequalities of rational expressions-dsc_0001-2-1-.jpg
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  2. #2
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    Re: inequalities of rational expressions

    \dfrac{5}{-4} < \dfrac{5}{4}

    If you were to cross-multiply, but keep the inequality as less than, you would get

    20 = 5(4) < 5(-4) = -20, which is clearly false.

    Now, if you have an expression like

    \dfrac{f(x)}{g(x)} < \dfrac{h(x)}{j(x)}, you must be certain of whether it is positive or negative. So, you can say the following:

    f(x)j(x) < g(x)h(x) if g(x)j(x)>0

    f(x)j(x) > g(x)h(x) if g(x)j(x)<0

    Example 1:

    \dfrac{5}{(x+1)^2} < \dfrac{7x}{5}

    Since 5(x+1)^2>0 for all x, you know 5(5) < 7x(x+1)^2

    Example 2:

    \dfrac{5}{-4} < \dfrac{7x^3}{1-(x+2)^2}

    Since -4[1-(x+2)^2] = 4[(x+2)^2-1] \ge 4[(0+2)^2-1] = 4(4-1) = 12>0, you know that 5[1-(x+2)^2] < -7x^3(-4)

    Example 3:

    \dfrac{5}{-4} < \dfrac{7x^3}{(x+1)^2}

    Since -4(x+1)^2 < 0 for all x, you know 5(x+1)^2 > 7x^3(-4) (note the inequality changed direction because of the single negative).
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