Math Is Fun Forum

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Modulus Inequality

The question gives the hint: "consider the graph of the curve with equation

", which I don't find very heplful at all - how would I go about drawing this curve?

I've approached it a slightly different way, but I'm not entirely sure it's a solid approach.

At this point we can just ignore the modulus on the right hand side, as that expression is positive for all real x anyway.

I've then sketched the two equations, and found where they meet:

And

I've then said that, because each of the solutions to these has a double root, the graphs only touch at those points but don't cross. This therefore means that the quadratic must either be greater than or equal to |2x+2|.

Is this sufficient?

Thanks.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Modulus Inequality

NB: The graph you’re asked to draw is the curve

shifted one unit to the left.

Last edited by JaneFairfax (2008-06-07 03:14:43)

Kurre

Member

Registered: 2006-07-18

Posts: 280

Re: Modulus Inequality

Now letting y=x+1 we get the desired inequality.

edit: about your siolution, instead of setting equality between the two functions to see where they meet, keep the inequality sign and consider the same two cases.

Last edited by Kurre (2008-06-07 05:52:36)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Modulus Inequality

Why, of course! I should have known that the AM–GM inequality for two variables is equivalent to (√a−√b)[sup]2[/sup] ≥ 0. Why did I have to do it the long way?
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Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: Modulus Inequality

Thanks both of you.