Question

If $|a|$ denotes the absolute value of an integer, then which of the following are correct?
1.$|ab|=|a||b|$
2. $|a+b|\le |a|+|b|$
3. $|a-b|\ge ||a|-|b||$
Select the correct answer using the code given below.

• A. 1 and 2 only
• B. 2 and 3 only
• C. 1 and 3 only
• D. 1, 2 and 3

Answer 1

The correct option is D 1, 2 and 3

Given $\left|a\right|$ is the absolute value of an integer,

From the definition,

$\left|a\right|=a$ if $a\ge 0$,

$=-a$ is $a\le 0$

$\therefore$ $\left|ab\right|=\left|a\right|\left|b\right|$ where $a,b$ are real numbers.

We know that from the triangle inequality sum of any two sides is always greater than the third side,

i.e.,$|a+b|\le \left|a\right|+\left|b\right|$,

We can also prove by considering

Absolute part of the difference between any two sides is always less than the third side,

$⟹|a-b|\ge |\left|a\right|-\left|b\right||$