explaining $|a+b|≤|a|+|b|$ in simple terms I'm struggling to get to grips with the Triangle Inequalities. The problem is I don't really understand what it means. This is what my lecturer has written in the notes:
$$
|a+b|≤|a|+|b|.
$$
First of all, I don't understand why it's less than or equal to.

If I plug in numbers, for example $a=2$ and $b=5$, I struggle to see how $|2+5|≤|2|+|5|$.

If someone could explain this to me I would be very grateful.
OK. I will try to explain why
$$
|a+b|\leq |a| + |b|
$$
is true in a somewhat intuitive/pictorial way.

First off, do you know what $|x|$ means for some number $x$? You probably know that $|x|$ means the absolute value of $x$, but what does that even mean? The absolute value of a number $x$ may be thought of as its "absolute" distance from zero on the real number line. For example, how far away is $3$ from $0$? This is clearly $3$ units away. What about $-4$? Well, this is just $4$ units away from $0$ but in a different direction (remember, the only thing that matters is absolute distance and distances are always positive). Thus, we get the following definition for the absolute value of a number $x$:
$$
|x|=
\begin{cases}
x &amp;\text{if} &amp; x\geq 0,\\
-x &amp;\text{if} &amp; x&lt;0.
\end{cases}
$$
This guarantees us that the distance from $x$ to $0$ will always be positive, regardless of the value of $x$. 

Phew. Okay. Now consider what $|a+b|\leq|a|+|b|$ means. What if $a$ is positive and $b$ is negative? What number will be larger: $|a+b|$ or $|a|+|b|$? Can you see how this might confirm the truth of the triangle inequality? 

If you need more explanation or want a general proof (although I think a general proof may confuse you more than help you at the moment), then let me know and I'll add it.
$|x|$ means the absolute value of $x$. This means that if the value of $x&lt;0$, then $|x|=-x$ otherwise $|x|=x$. In simple words, $|x|$ is formed by removing the minus sign from $x$, if any. So $|4|=4$ and $|-3|=3$

The equation states an obvious fact:
$$|a+b|\leq|a|+|b|$$

If $a$ and $b$ are both positive, we have $|a+b|=|a|+|b|$, else we have $|a+b|&lt;|a|+|b|$

Try putting these values, and you should be able to figuring it out: $(2,5)$, $(2,-5)$, $(-2,-5)$

What this has to do with a triangle is unclear to me, though.