### Inequalities - Two more questions

1. (|x| - 2) ( x + 5) < 0. What is the range of values x can take?

2. a and b are roots of the equation x^2 - px + 12 = 0. If the difference between the roots is at least 12, what is the range of values p can take?

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**Solutions to the above questions**

**1. (|x| - 2) ( x + 5) < 0 -**

This can be true in two scenarios

Scenario I - (|x| - 2) < 0 and ( x + 5) > 0

Or |x| < 2 and x > -5.This gives us the range (-2,2)

Scenario II - (|x| - 2) > 0 and ( x + 5) < 0

Or |x| > 2 and x < -5. This gives us the range (-Infinity, -2)

So, the overall range is (-infinity, -2) or (-2,2)

**2. a and b are roots of the equation x^2 - px + 12 = 0. If the difference between the roots is at least 12, what is the range of values p can take?**

The roots are a and b

a + b = p ab = 12

(a + b )^2 = p^2

(a -b)^2 = (a + b ) ^2 - 4ab

=> (a-b) ^2 = p^2 - 12*4 = p^2 - 48

If |a-b|

__>__12 { Difference between the roots is at least 12}

then, (a-b)^2

__>__144

p^2 - 48

__>__144

p^2

__>__192

P

__>__8sqrt(3) or P

__<__-8 sqrt(3)