Question

The equation $ \left| {\left| {x - 1} \right| + a} \right| = 4 $ can have real solutions for x if a belongs to the interval
A. $ ( - \infty ,4] $
B. $ ( - \infty , - 4] $
C. $ [4, + \infty ) $
D. $ [ - 4,4] $

Answer 1

Hint: The modulus function provides the absolute value of a number or variable. It always provides a positive value of the number even if it’s negative. Real numbers are those which can be represented on a continuous line, they include both rational and irrational numbers. Using these two definitions solve the given question.

Complete step-by-step answer:
We are given that $ \left| {\left| {x - 1} \right| + a} \right| = 4 $
Removing the modulus, we get - $ \left| {x - 1} \right| + a = \pm 4 $
 $ \Rightarrow \left| {x - 1} \right| = 4 - a $ or $ \left| {x - 1} \right| = - 4 - a $
For x to be real $ 4 - a \geqslant 0 $ or $ - 4 - a \geqslant 0 $
 $ \Rightarrow a \leqslant 4 $ or $ a \leqslant - 4 $
Thus a lies between $ ( - \infty ,4] $ or $ ( - \infty , - 4] $
As the interval $ ( - \infty , - 4] $ is included in the interval $ ( - \infty ,4] $ , thus a belongs to the interval $ ( - \infty ,4] $ for the equation to have real solutions for x.
So, the correct answer is “Option A”.

Additional Information.:
There are different types of numbers in mathematics as follows:
Natural numbers – these are positive counting numbers. $ (1,2,3....) $
Whole numbers – The set of natural numbers including zero. $ (0,1,2,3....) $
Integers – These are negative counting numbers plus the whole numbers. $ (.... - 2, - 1,0,1,2....) $
Rational numbers – These are all the fractions where the numerator and denominator both are integers but the denominator cannot be zero. $ (\dfrac{1}{2},\dfrac{3}{8},\dfrac{{ - 4}}{3}.....) $
Irrational numbers – They cannot be expressed as a simple fraction, they have neither terminating nor recurring decimal extension. $ (\pi ,\sqrt 2 ....) $


Note: All the numbers that can be written as a decimal are called real numbers. It includes all the types of numbers like integers, rational numbers and irrational numbers. The other type of numbers is called imaginary numbers. The interval $ [ - 4,4] $ is also included in the interval $ ( - \infty ,4] $ so we consider it as the most suitable answer.