# Quantitative Reasoning : Inequalities

## Question

For what values of 'x' is the function

defined in the real domain?

- -10 < x < 4
- –4 < x < 10
- x does not lie between the closed interval –10 and 4
- x does not lie between the open interval –4 and 10

#### Correct Answer

Choice (4). x does not lie between the open interval –4 and 10

#### Explanatory Answers

The function

is defined in the real domain only when x

^{2} – 6x – 40

__>__ 0.

When x

^{2} – 6x – 40 is < 0, the function will be imaginary.

Now let us find out the range of values for which x

^{2} – 6x – 40

__>__ 0.

Factorizing the quadratic expression, we get (x – 10)(x + 4)

__>__ 0

This expression (x – 10)(x + 4) will be greater than or equal to 0 when both (x – 10) and (x + 4) are greater than or equal to 0 or when both (x - 10) and (x + 4) are less than or equal 0.

## Case 1:

When both (x – 10) and (x + 4) are greater than or equal to 0.

x

__>__ 10 and x

__>__ - 4 => when x

__>__ 10 it will be greater than –4.

Therefore, it will suffice to say that x

__>__ 10

## Case 2:

When both (x - 10) and (x + 4) are less than or equal to 0.

i.e. x

__<__ 10 and x

__<__ -4 => when x

__<__ -4, it will less than 10.

Therefore, it will suffice to say that x

__<__ -4

Hence, the range in which the given function will be defined in the real domain will be when x does not lie between –4 and 10.

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