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Math Help - Quadratic Inequality (Why doesn't this work?)

  1. #1
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    Quadratic Inequality (Why doesn't this work?)

    Q. What is the solution of x^2 + 2x - 24 > 0?
    Solution: X < -6, X > 4
    Things are more clear on a graph but it doesn't work when I work it through.


    My method gave me the solution:

    X > -6, X > 4. What does it mean? What did I do wrong? What did I imply?


    What I did:
    1. Factored the polynomial

    (x + 6)(x - 4) > 0

    2. Divide each side by (x + 6)

    (x - 4) > /frac {0}{x + 6}

    3. x is greater than 4 to satisfy the equation.

    (x - 4) > 0

    4. To find the other constraint, I divide each side by (x - 4) which leaves me:

    (x + 6 ) > 0

    5. To satisfy the equation, x > -6
    (-7 + 6 = -1)
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  2. #2
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    Quote Originally Posted by Masterthief1324 View Post
    What I did:
    1. Factored the polynomial

    (x + 6)(x - 4) > 0
    Excellent.

    2. Divide each side by (x + 6)

    (x - 4) > /frac {0}{x + 6}
    Not excellent. This is very, very bad. What if x = -6? This just blew up! What if x < -6? You forgot to switch the inequality. Never do this. After factoring, you must think about what you are doing. Guessing will not do.

    If a product is positive, what do you know about the factors?

    1) Both are positive, or
    2) Both are negative.

    1) x > 4 both are positive.
    2) x < -6 both are negative.
    3) -6 < x < 4 one of each and the product is negative.
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  3. #3
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    Quote Originally Posted by Masterthief1324 View Post
    Q. What is the solution of x^2 + 2x - 24 > 0?
    Solution: X < -6, X > 4
    Things are more clear on a graph but it doesn't work when I work it through.


    My method gave me the solution:

    X > -6, X > 4. What does it mean? What did I do wrong? What did I imply?


    What I did:
    1. Factored the polynomial

    (x + 6)(x - 4) > 0

    2. Divide each side by (x + 6)

    (x - 4) > /frac {0}{x + 6}

    3. x is greater than 4 to satisfy the equation.

    (x - 4) > 0

    4. To find the other constraint, I divide each side by (x - 4) which leaves me:

    (x + 6 ) > 0

    5. To satisfy the equation, x > -6
    (-7 + 6 = -1)
    Don't forget if you divide by a negative number, you have to switch the direction of the inequality. Since you already found that one solution is for x > 4, for x < 4 (x - 4) will be less than 0 and therefore you have to switch the direction. So your step 4 should be

    (x + 6 ) < 0

    giving x < -6, which is what you want.
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  4. #4
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    As lgstarn said, if you multiply or divide an inequality by a negative number, you must reverse the direction of the inequality.

    The difficulty with solving such inequalities is that you don't know x beforehand and so don't know if the expression you are dividing by is positive or negative.

    For factorable polynomials, for example (x+6)(x-4), you can do what TKHunny suggested: a product of two numbers is positive if and only if both factors are of the same sign, negative if and only if they are of different sign. (x+6)(x-4)> 0 if and only if x+6> 0 and x-4> 0 or if x+6< 0 and x- 4< 0. x+6> 0 for x>-6 and x-4> 0 for x> 4. In order for both of those to be true, you must have x> 4. x+6< 0 for x< -6 and x-4< 0 for x< 4. In order for both of those to be true, you must have x< -6.

    Here's another technique that works even for non-polynomials. A continuous function can change sign only where it is equal to 0 (and rational functions, fractions, which are not continuous, where the denominator is equal to 0). You can check just one point in each interval between such points to see if the expression is ">" or "< ".

    Start by solving the equation (x+6)(x-4)= 0 for x= -6 and x= 4, of course. Now check the value in the intervals, x< -6, -6< x< 4, and 4< x.
    If x= -7< -6, then (x+6)(x-4)= (-7+6)(-7-4)= (-1)(-11)= 11> 0. If x= 0, which is between -6 and 4, (x+6)(x-4)= (0+6)(0-4)= (6)(-4)= -24< 0. If x= 5> 4, then (x+6)(x-4)= (5+6)(5-4)= (11)(1)= 11> 0.

    Again, we find that (x+6)(x-4)> 0 for x< -6 or x> 4.
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  5. #5
    Junior Member SuperCalculus's Avatar
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    I also generally find it helpful, if you're not sure which inequality sign to use in your answer, to sketch the graph, look at the inequality side in the original equation, and from there simply logically figure it out. For example, if you'd sketched the graph for the equation in this graph after factorising, you'd know that the left side is greater than 0, and the curve crosses the x axis at -6 and 4. As it's GREATER than 0, it's the bit above the x-axis, which makes it clear that x<-6 and x>4
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