Question

Fill in the blanks with \[ > \], \[ < \] or \[ = \] sign: -
\[ - 45 - \left( { - 11} \right)\] ? \[57 + \left( { - 4} \right)\]
(a) \[ > \]
(b) \[ < \]
(c) Cannot be determined
(d) None of these

Answer 1

Hint: The given problem revolves around the concepts of algebraic solution, representing the greater than ‘\[ > \]’, lower than ‘\[ < \]’ and equal to ‘\[ = \]’ sign respectively. As a result, to get the desired sign, solving the given expression of both sides that is L.H.S. as well as R.H.S. respectively, the desired condition is obtained.

Complete answer:
Since, we have given that
\[ - 45 - \left( { - 11} \right)\] ? \[57 + \left( { - 4} \right)\]
Since, given that we have to determine the condition that is which side is greater or lower or both are same that is left hand side ( L.H.S. ) or right hand side ( R.H.S. ) using the mathematical signs that is ‘\[ > \]’, ‘\[ < \]’ or ‘\[ = \]’ respectively.
Therefore, solving both the sides respectively so as to find the desire value for the determination of the required condition, we get
Hence,
Considering, the left hand side (L.H.S.) of the given expression that is \[ - 45 - \left( { - 11} \right)\], we get
As a result, solving it mathematically, we get
\[ - 45 - \left( { - 11} \right) = - 45 + 11\]
(Where, ‘minus’ & ‘minus’ becomes ‘plus’)
\[\therefore - 45 - \left( { - 11} \right) = - 34\] … (i)
Similarly,
Considering the right hand side (R.H.S.) of the given expression that is \[57 + \left( { - 4} \right)\], we get
\[57 + \left( { - 4} \right) = 57 - 4\]
(Where, ‘plus’ & ‘minus’ becomes ‘minus’)
\[\therefore 57 + \left( { - 4} \right) = 53\] … (ii)
As a result,
From (i) and (ii), it is observed that
\[53\] is greater than \[ - 34\] as per as the number system is concerned.
Hence, mathematically it can also be represented as
\[\left( {53} \right) > \left( { - 34} \right)\] i.e.
The given expression seems,
\[\left[ {57 + \left( { - 4} \right) = 53} \right] > \left[ { - 45 - \left( { - 11} \right) = - 34} \right]\]
By associative law that is, if \[a = b\] then \[b = a\],
 \[\left[ { - 45 - \left( { - 11} \right) = - 34} \right] < \left[ {57 + \left( { - 4} \right) = 53} \right]\]
\[\therefore \Rightarrow \]Option (b) is correct.

Note:
One must be able to know basic properties while solving such (silly) expressions/equations such as in multiplication minus, minus becomes plus; plus, minus becomes minus, etc. as introduced above. Also, remember all the rules of indices like \[{a^m} \times {a^n} = {a^{m + n}}\], \[{\left( {{a^m}} \right)^n} = {a^{mn}}\], etc. properties or, laws which includes (in terms of) in mathematics such as Associative law \[a + \left( {b + c} \right) = \left( {a + b} \right) + c\], Distributive law \[a \times \left( {b + c} \right) = a \times b + a \times c\], commutative law \[a + b = b + a\], so as to be sure of our final answer.