The set of all values of `x` for which
`|f(x) + g(x)| lt |f(x)| + |g(x)|` is true if `f(x) = x − 3` and `g(x) = 4 − x` is given by

Options

A. `R`
B. ` R-(3, 4)`
C. `R − [3, 4]`
D. none of these

Answer

B

Explanation

(c) The given inequality is equilvalent to
`1 lt |x − 3| + |4 − x|`
i.e.`1 lt |x − 3| + |x − 4|.`
For `x lt 3`, we have
`3 − x + 4 − x = 7 − 2x gt 1`...... (1)
i.e., `x lt 3`, which is true in the domain.
For `3 lt= x lt 4 `(1) gives
`x − 3 + 4 − x = 1 gt 1.`......(2)
For `4 lt= x` we have
`2x − 7 gt 1 implies x gt 4`......(3)
From (1) and (3), we have the solution of the inequality as `x lt 3` or `x gt 4` i.e `x in ] −oo, 3 [ ∪ ] 4, oo [` which is equal to `R − [3, 4].`