### Instruction: Find the least integer value of $x$ for which $2|x| + 7 < 17$.### Response: I need to find the smallest integer that makes the inequality true.
I notice that the inequality involves absolute value, so I need to be careful with the sign of $x$.
To simplify the inequality, I can subtract 7 from both sides, getting $2|x| < 10$.
Then I can divide both sides by 2, getting $|x| < 5$.
This means that $x$ can be any integer between -5 and 5, excluding the endpoints.