### Inequalities of a quadratic function for real values of x. Inequality Problems. Ascent TANCET, CAT, PGSEM Correspondence course

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# Quantitative Reasoning : Inequalities

## Question

For what values of 'x' is the function defined in the real domain?
1. -10 < x < 4
2. 4 < x < 10
3. x does not lie between the closed interval 10 and 4
4. x does not lie between the open interval 4 and 10

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

The function is defined in the real domain only when x2  6x  40 > 0.

When x2  6x  40 is < 0, the function will be imaginary.

Now let us find out the range of values for which x2  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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