Absolute Value (Hard)

Late AMC 10/12 through early AIME — dense computation.

Absolute Value: core idea

Inequalities.

Solve $|2x-5|=9$ ; find the sum of solutions.

Hint: Split an absolute value by sign, or interpret it as distance. Answer: 5.

1. Find $\min_x(|x-3|+|x-11|)$ .
2. How many integers satisfy $|x-4|<7$ ?
3. Solve $|x+2|=|3x-6|$ ; find product of solutions.
4. Find $\max_{-3\le x\le7}|x-2|$ .
5. $y=|x-a|$ passes through $(5,7)$ ; its positive vertex is $(a,0)$ . Find $a$ .
6. Find $\min_x(|x|+|x-4|+|x-10|)$ .
7. How many real solutions has $|x^2-5x|=6$ ?
8. Find sum of integers satisfying $|2x+1|\le7$ .
9. For $x\ne0$ , find $\min(|x|+9/|x|)$ .