Question

The set of values of $x$, for which $\mid x - 1\mid + \mid x + 1\mid < 4$ always holds true, is ?

Answer 1

Here we are given an expression and we need to find the set of the values of x. For this, we know that we will get two values of x when we open the modulus. Then, we will use different cases to get the set of values of x. And combining them all, we will get the final output.

Complete step by step answer:
Given that, $\mid x - 1\mid + \mid x + 1\mid < 4$. When we open the mode we will get two values of x, x = -1, 1
Case 1: $x < \- 1$
Applying this value of x in above expression given, we will get,
$\Rightarrow - (x - 1) + ( - (x + 1)) < 4$
Opening the brackets, we will get,
$\Rightarrow - x + 1 + ( - x - 1) < 4$
$\Rightarrow - x + 1 - x - 1 < 4$
On evaluating this, we will get,
$\Rightarrow - 2x < 4$
On transposing this, move the LHS term to RHS term, we will get,
$\Rightarrow 0 < 4 + 2x$
Again using the transposing method, we will move RHS term i.e. 4 to LHS, we will get,
$\Rightarrow - 4 < 2x$
Moving the minus sign from LHS to RHS, we will get,
$\Rightarrow 4 < \- 2x$
$\Rightarrow - 4 < 2x$
$\Rightarrow 2x > - 4$
$\Rightarrow x > - 2$
$\therefore x \in ( - 2, - 1)$

Case 2: $- 1 \leqslant x < 1$
Applying this value of x in above expression given, we will get,
$\Rightarrow - (x - 1) + (x + 1) < 4$
Opening the brackets, we will get,
$\Rightarrow - x + 1 + x + 1 < 4$
On evaluating this, we will get,
$\Rightarrow 2 < 4$
$\therefore x \in [ - 1, - 1)$

Case 3: $x \geqslant 1$
Applying this value of x in above expression given, we will get,
$\Rightarrow (x - 1) + (x + 1) < 4$
Opening the brackets, we will get,
$\Rightarrow x - 1 + x + 1 < 4$
On evaluating this, we will get,
$\Rightarrow 2x < 4$
$\Rightarrow x < 2$
$\therefore x \in [1,2)$
Combining all the three cases, the value of x belongs from -2 to 2 i.e. $x \in ( - 2,2)$

Hence, the set of values of x is $x \in ( - 2,2)$.

Note: The magnitude of a number (also called its absolute value) is its distance from zero. For example, the magnitude of 3 is 3 and the magnitude of -3 is 3. The magnitude of a vector is its length (ignoring direction). In short, the magnitude is the number itself. Also, the magnitude of a complex number is its distance from the origin in the complex plane.