If |a| denotes the absolute value of an integer, then which of the following are correct? 1.|ab|=|a||b| 2. |a+b|≤|a|+|b| 3. |a−b|≥||a|−|b|| Select the correct answer using the code given below.
A
1 and 2 only
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B
2 and 3 only
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C
1 and 3 only
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D
1, 2 and 3
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Solution
The correct option is D 1, 2 and 3 Given |a| is the absolute value of an integer,
From the definition,
|a|=a if a≥0,
=−a is a≤0
∴|ab|=|a||b| 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|≤|a|+|b|,
We can also prove by considering
Absolute part of the difference between any two sides is always less than the third side,