Triangle Inequality: Solve |(a+b)-13| < 1

[physstudent1]

Homework Statement

Show that if |a-5| < 1/2 and |b-8| < 1/2 then |(a+b)-13| < 1. Hint: use the triangle inequality.

Homework Equations

Triangle Inequality:

|a+b| <= |a|+|b|

The Attempt at a Solution

I really don't know how to use the triangle inequality so I was hoping someone could clear up for me exactly how it is used my book doesn't really make it clear it just states what it is which is what I have stated above. I understand why it is true, I just do not understand how you would use it in a problem. I plugged the first parts into it to get |(a-5)+(b-8)| <= |a-5| + |b-8| I'm not really sure how to simplify this though it should simplify to |a+b-13| but I can't get that everything is just canceling out for me.

 

[cristo]

Apply the triangle inequality to (a-5) and (b-8). Then |(a-5)+(b-8)|=|a+b-13|$\leq$|a-5|+|b-8|<1/2+1/2=1

The first inequality is the triangle inequality, and the second is from the original information.

 

[genneth]

You're almost there: expand out the brackets on the left hand, and add a further inequality to the right, using the information that you've been given, but which you've not yet used.

 

[physstudent1]

k so I ended up getting |a+b-13|<= |a+5 + |b-8| < 1 thanks a lot guys.