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    Hi, I'm struggling with how to solve inequalities, involving modulus (in core 3). In particular, how do you tell whether it's X > a, X < a, a < X < b? (Basically how do you determine which sign the X has?)

    I've attached a worked example (see photo), which I can follow up to 3x^2 - 8x - 3 > 0, but I can't understand where they've got X > 3, X < -1/3 (eg. Why is it not X > -1/3, if that's the way around the the sign has been all thoughout before?)

    So simply, does anyone have a generalised procedure, for how you'd determine whether its < or >, for any modulus inequality?

    Thanks


    Note: ignore the doodles on the picture, that's just my working



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    (Original post by JohnnyDavidson)
    Hi, I'm struggling with how to solve inequalities, involving modulus (in core 3). In particular, how do you tell whether it's X > a, X < a, a < X < b? (Basically how do you determine which sign the X has?)

    I've attached a worked example (see photo), which I can follow up to 3x^2 - 8x - 3 > 0, but I can't understand where they've got X > 3, X < -1/3 (eg. Why is it not X > -1/3, if that's the way around the the sign has been all thoughout before?)

    So simply, does anyone have a generalised procedure, for how you'd determine whether its < or >, for any modulus inequality?

    Thanks


    Note: ignore the doodles on the picture, that's just my working



    Name:  image.jpg
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    If you sketch the graph of y = (3x+1)(x-3) and shade the bits where y>0 that should tell you.

    If you're also asking in general about the mod function, how mod(x) > a implies that x<-a and x>a and mod(x) < b implies that -b<x<a for some numbers a and b, then let me know.
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    (Original post by SeanFM)
    If you sketch the graph of y = (3x+1)(x-3) and shade the bits where y>0 that should tell you.

    If you're also asking in general about the mod function, how mod(x) > a implies that x<-a and x>a and mod(x) < b implies that -b<x<a for some numbers a and b, then let me know.
    Thanks, do you have any advice for how to draw the graph for this equation? It seems quite a complicated line, I can tell it will have a U X squared shape,not sure where I would start with drawing it.

    Ah, so in this case, since it's 3x^2 - 8x - 3 > 0, as there's a > sign, there must be an X >, and an X <?

    whereas if it was it's 3x^2 - 8x - 3 < 0, it would be < X < ?
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    i always do a graph ?
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    (Original post by JohnnyDavidson)
    Thanks, do you have any advice for how to draw the graph for this equation? It seems quite a complicated line, I can tell it will have a U X squared shape,not sure where I would start with drawing it.

    Ah, so in this case, since it's 3x^2 - 8x - 3 > 0, as there's a > sign, there must be an X >, and an X <?

    whereas if it was it's 3x^2 - 8x - 3 < 0, it would be < X < ?
    Look at the roots of the equation (where y=0) and use what you know about the shape of +x^2 and -x^2 graphs and you're pretty much done.

    The modx thing is slightly different and doesn't have anything to do with the quadratic.

    If mod(x) > 5 for example, then x is either greater than 5 or less than -5. You can see why by testing 6, 4, -4 and -6.

    And if mod(x) < 8, then x is between -8 and 8. You can see this by testing 9, 7, -7 and -9.
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    (Original post by SeanFM)
    Look at the roots of the equation (where y=0) and use what you know about the shape of +x^2 and -x^2 graphs and you're pretty much done.

    The modx thing is slightly different and doesn't have anything to do with the quadratic.

    If mod(x) > 5 for example, then x is either greater than 5 or less than -5. You can see why by testing 6, 4, -4 and -6.

    And if mod(x) < 8, then x is between -8 and 8. You can see this by testing 9, 7, -7 and -9.
    Ah ok, so you don't have to draw the graph out, just recognise from the shape of an x^2?

    eg. X^2 > 0, will be above a certain value, and below a certain value, so will be X < and X >

    Whereas X^2 < 0, will always be between 2 values, so < X <



    And thanks for the mod > X, I didn't realise it was different to mod < X
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    (Original post by JohnnyDavidson)
    Ah ok, so you don't have to draw the graph out, just recognise from the shape of an x^2?

    eg. X^2 > 0, will be above a certain value, and below a certain value, so will be X < and X >

    Whereas X^2 < 0, will always be between 2 values, so < X <



    And thanks for the mod > X, I didn't realise it was different to mod < X
    Always draw the graph. It helps .
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    (Original post by JohnnyDavidson)
    Ah ok, so you don't have to draw the graph out, just recognise from the shape of an x^2?

    eg. X^2 > 0, will be above a certain value, and below a certain value, so will be X < and X >

    Whereas X^2 < 0, will always be between 2 values, so < X <



    And thanks for the mod > X, I didn't realise it was different to mod < X
    Yes, you've summed it up there. But I would sketch it and see that this is true rather than memorising it.

    Eg x^2 - 3x + 2 has roots x=2 and x=1. The graph is a u shape, so when x is less than 1 y is positive, and between 1 and 2 y is negative, and after 2 it is positive again (thanks to the U shape). So you can see when y>0 and y<0.
 
 
 
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