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# C3 Solving inequalities - Modulus watch

1. 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

2. (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

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.
3. (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 < ?
4. i always do a graph ?
5. (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.
6. (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
7. (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 .
8. (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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