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Math Help - Solve the inequality

  1. #1
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    Solve the inequality

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  2. #2
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    You have \displaystyle \frac{3}{a + 1} \geq 3.

    First, note that \displaystyle a + 1 \neq 0 \implies a \neq -1.

    In order to solve for \displaystyle a, you need to consider two cases, since multiplying or dividing by a negative number reverses the inequality sign...


    Case 1: \displaystyle a + 1 < 0 \implies a < -1.

    Then \displaystyle \frac{3}{a + 1} \geq 3

    \displaystyle 3 \leq 3(a + 1)

    \displaystyle 1 \leq a + 1

    \displaystyle 0 \leq a.

    So in this case, you are told \displaystyle a < -1 and \displaystyle a \geq 0. Since there are not any values of \displaystyle a that satisfy both conditions, there are not any solutions in this case.


    Case 2: \displaystyle a + 1 > 0 \implies a > -1.

    \displaystyle \frac{3}{a + 1} \geq 3

    \displaystyle 3 \geq 3(a + 1)

    \displaystyle 1 \geq a + 1

    \displaystyle 0 \geq a.

    So in this case, the conditions are \displaystyle a >-1 and \displaystyle a \leq 0.

    Therefore, the solution in this case is \displaystyle -1 < a \leq 0.


    So the solution of your inequality is \displaystyle -1 < a \leq 0.
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  3. #3
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    Wow, that is a big post, what have you tried?
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  4. #4
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    Quote Originally Posted by pickslides View Post
    Wow, that is a big post, what have you tried?
    Shouting it from the roof tops!
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