Absolute Value Inequalities

Absolute Value Inequalities: Level 3 Challenges


1x1+x12\left | 1 - \frac{|x|}{1 + |x|} \right| \geq \frac{1}{2}

The solution set to this inequality is axb.a \le x \le b. What is ba?b-a?

How many integers xx satisfy the inequality x2000+x9999 |x-2000|+|x| \leq 9999?

How many ordered pairs of integers (x,y) (x,y) are there that satisfy x+y10 |x| + |y| \leq 10 ?

Consider all monic polynomials f(x)=x2+bx+cf(x) = x^2 + bx + c , where bb and cc are real numbers. What is the minimum value of NN, where

N=maxx[10,10]f(x)? N = \max_{x \in [-10,10]} \vert f(x) \vert?

Details and assumptions

The last equation states: "The maximum value of the absolute value of f(x)f(x) , as xx ranges from 10-10 to 1010 inclusive".

2=x2+x4\large 2=|x-2|+|x-4|

Find the solution set of the above equation.


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