Algebra

Absolute Value Inequalities

Absolute Value Inequalities - Problem Solving

         

For all ordered triples of real values (a,b,c) (a, b, c) , which of these numbers is larger?

If x,y,z x, y, z are non-zero real, what is the range of

f(x,y,z)=x+yx+y+y+zy+z+z+xz+x? f(x, y, z) = \frac{ |x+y| } { |x| + |y| } + \frac{ |y+z| } { |y| + |z| } + \frac{ |z+x| } { |z| + |x| } ?

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[22,50]f(x)? N = \max_{x \in [-22, 50]} \vert f(x) \vert?

Consider all monic polynomials of the form fa(x)=x+af_a(x) = x + a . As aa ranges over all real numbers, what is the minimum value of NaN_a, where

Na=maxx[13,39]fa(x)? N_a = \max_{x \in [-13, 39] } \lvert f_a(x) \rvert ?

If the solution to the inequality x+a8 \lvert x+ a \rvert \leq 8 is 7x23, 7 \leq x \leq 23, what is a?a?

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