How to use Triangle inequality to prove $|(x+y)-5| < 0.05$ when $|x-2| < 0.01$ and $|y-3| < 0.04$ It's the first day of calculus, and it's been almost a year since I've been in college algebra, and really stuck on the following homework question:
"Suppose that $| x - 2| < 0.01$ and $| y - 3 | < 0.04$. Use the
Triangle Inequality to show that $| (x + y) — 5 | < 0.05$."
I don't even know how to start this one, and I can't find anything remotely similar on google. I know that the triangle inequality is $|x+y| = |x| + |y|$, but I don't know how it relates to this question.
I would actually prefer if an alternative answer could be given, and allowed to work it out myself if possible, as I feel like I need to learn this
EDIT: So I am writing this the next day, apparently the reason I was so confused was that we hadn't gone over it in class yet -_- Thanks to everyone who helped though
 A: $$|x+y-5|=\underbrace{|(x-2)+(y-3)|\le|x-2|+|y-3|}_{\text{triangle inequality}}<0.05$$
A: This is just straightforward. You get
$$ \vert x + y - 5 \vert = \vert x - 2 + y - 3 \vert \leq \vert x - 2 \vert + \vert y - 3 \vert < 0.01 + 0.04 = 0.05$$
using the triangle inequality. I hope it helps you :)
A: One thing about the problem that was given to you that could confuse you
is that it uses the same symbols $x$ and $y$ that are in the
Triangle Equality (at least in the version of that fact that you've seen).
So rename the variables in the Triangle Inequality.
They're just arbitrary names for anything you could plug into the formula,
after all.
You can just as easily write
$$
\lvert a + b \rvert \leq \lvert a \rvert + \lvert b \rvert
$$
(but do remember it's an inequality, specifically $\leq$, not $=$).
Now you have a known fact with three things in absolute values, and
you have three other things in absolute values that you've been asked to
say something about. So let's try matching up the three things you
were asked about with the three things you know about.
For example, you could try $a = (x + y) - 5$ and $b = x -2$.
Then $a + b = \ldots$ ?
As you can see if you try that, it wasn't very useful.
So try a different way to match $a$ and $b$ with $x - 2,$
$y - 3,$ or $(x + y) - 5,$ and see what you get for $a+b.$
