Algebra
Absolute Value Inequalities
Solve for \(x:\) \[ \lvert x + 3 \rvert + \lvert x-3 \rvert > 16 .\]
How many integers \(x\) satisfy the inequality \[3\lvert x-1 \rvert + 4\lvert x+1 \rvert \le 45?\]
If the solution to the inequality \[7\lvert x+6 \rvert \le 6\lvert x+7 \rvert\] is \(\alpha \le x \le \beta,\) what is \(13(\alpha+\beta)?\)
How many integers \(x\) satisfy the inequality \[\big\lvert \lvert x-1 \rvert -5 \big\rvert \le 13?\]
What is the range of \(x\) that satisfies the inequality \[\lvert x+1 \rvert + \sqrt{x^2-4x+4}<x+82?\]