Algebra

Absolute Value Inequalities

Absolute Value Inequalities - 3 or More Linear Terms

         

What is the smallest positive integer nn, such that there exists nn real numbers {xi}i=1n \{ x_i \} _{i=1}^n which satisfy xi<1 | x_i | < 1 and

x1+x2+x3++xn=165+x1+x2++xn? | x_1| + |x_2| + |x_3| + \ldots + |x_n| = 165 + |x_1 + x_2 + \ldots + x_n| ?

As xx ranges over all real numbers, what is the smallest value of

f(x)=x165+x247+x316? f(x) = |x- 165 | + |x-247 | + |x- 316 | ?

Solve for x:x: x+162x16+x>4. \lvert x+16 \rvert - 2\lvert x - 16 \rvert + \lvert x \rvert > 4 .

Six children are standing along the xx-axis at points (0,0)(0,0), (30,0)(30,0), (87,0)(87,0), (142,0)(142,0), (237,0)(237,0), (504,0)(504,0). The children decide to meet at some point along the xx-axis. What is the minimum total distance the children must walk in order to meet?

As xx ranges over all real values, what is the minimum value of f(x)=x333+x631+x856 f(x)=|x-333|+|x-631| + |x-856| ?

Details and assumptions

The notation | \cdot | denotes the absolute value. The function is given by x={xx0xx<0 |x | = \begin{cases} x & x \geq 0 \\ -x & x < 0 \\ \end{cases} For example, 3=3,2=2 |3| = 3, |-2| = 2 .

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