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Math Help - Solving an inequality

  1. #1
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    Solving an inequality

    Can anyone tell me how to solve an inequality like this?

    x^2+x-6<0.

    I tried factorising it like this:

    (x+3)(x-2)<0

    But I don't see how I can solve it from there. I'd just like to know the method please, so I can do it myself thank you.
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  2. #2
    MHF Contributor
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    Inequalities

    Hello Lupin
    Quote Originally Posted by Lupin View Post
    Can anyone tell me how to solve an inequality like this?

    x^2+x-6<0.

    I tried factorising it like this:

    (x+3)(x-2)<0

    But I don't see how I can solve it from there. I'd just like to know the method please, so I can do it myself thank you.
    First, you need to find the values of x
    that make the function zero. Since you've factorised it, that's now very easy: x = -3 and x = 2.

    Next, imagine the number line with these numbers
    -3 and 2 marked, and the number x moving from left to right along the line. When it's right over on the left-hand side, x < -3. When it's somewhere in the middle -3<x<2, and when it's going off at the right-hand end, x > 2. So, look at the signs of your factors (x+3) and (x-2) in each of these three ranges of values of x. For instance:

    When x < -3 (e.g. x = -10), (x+3) is negative and (x-2) is also negative. And a negative times a negative equals a positive. So (x+3)(x-2) is positive. Which is not what we want: we want values of x that make (x+3)(x-2)<0.

    Now look at the other two ranges: -3<x<2 and x>2, and see what happens to the signs in each case.

    I hope you can finish it now.

    Grandad

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  3. #3
    Math Engineering Student
    Krizalid's Avatar
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    x^2+x-6<0\implies4x^2+4x-24<0\implies(2x+1)^2<25, then -5<2x+1<5 so -3<x<2.
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  4. #4
    Senior Member pankaj's Avatar
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    Here is a gift for you Lupin
    Solve for x:
    (i) (x-1)(x-2)(x-3)(x+5)<0

    (ii) (x-2)^2 (x+4)^3(x-7)(x-11)>0

    I hope you enjoy working on them.
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  5. #5
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    Hello, Lupin!


    Can anyone tell me how to solve an inequality like this?

    x^2+x-6\:<\:0.

    I tried factorising it like this: . (x+3)(x-2)\:<\:0

    But I don't see how I can solve it from there.

    We have a parabola: . y \:=\:x^2 + x - 6 \:=\:(x+3)(x-2)

    It opens upward and has x-intercepts -3 and 2.
    So it looks like this:
    Code:
                    |
          *         |      *
                    |
           *        |    *
        - - * - - - + - * - - - - -
           -3 *     | * 2
                  * |
                    |

    And they are asking: When is the function negative?

    . . That is: When is the parabola below the x-axis?

    Got it?

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  6. #6
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    Thanks for all the help everyone. I totally understand this now. As for Pankaj's questions:

    (i) -5<x<1 and -2<x<3

    (ii) -4<x<7

    I hope they're right, and thanks for the questions.
    Last edited by Lupin; January 19th 2009 at 06:16 PM.
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  7. #7
    Senior Member pankaj's Avatar
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    Quote Originally Posted by Lupin View Post
    Thanks for all the help everyone. I totally understand this now. As for Pankaj's questions:

    (i) -5<x<1 and -2<x<3

    (ii) -4<x<7

    I hope they're right, and thanks for the questions.
    (i)There appears to be a typo .In place of -2 you should have written 2.

    (ii)There is a very simple way out to do such questions.Plot all the numbers on the number line where the factors become zero.Here we see that -4,2,7,11 are the numbers.
    The number line gets split into 5 parts/intervals.
    (-) .........(+) .........(+)............... (-)................ (+)
    --------------------------------------------------------------Number line
    ........-4 ..........2....... .......7 ...............11...........

    Starting from the rightmost interval start putting the (+) and (-) signs alternatively.But remember sign will not change across that number whose corresponding factor has even power,2 in this case.And you get the desired intervals which are x\in(-4,2)\cup(2,7)\cup(11,\infty)
    Remember to exclude 2 since at x=2 LHS becomes 0.Your initial question can be answered in the same manner
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  8. #8
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    Post

    read this doc file
    Attached Files Attached Files
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