To find the value of G, we need to evaluate the expression 1-|x+1|-|x+2|-...-|x+10|.

Let's break it down step by step:

1. Start with the expression 1-|x+1|-|x+2|-...-|x+10|.
2. The absolute value of a number is always non-negative, so each term |x+n| will be non-negative.
3. Since we are subtracting each term, we want to make each term as large as possible to minimize the overall value.
4. To make each term as large as possible, we want to make x+n as small as possible (negative).
5. The smallest value of x+n occurs when x = -10.
6. Plugging in x = -10, we get: 1-|-10+1|-|-10+2|-...-|-10+10|.
7. Simplifying each term, we get: 1-|-9|-|-8|-...-|0|.
8. The absolute value of a negative number is the positive value of that number, so we have: 1-9-8-...-0.
9. Evaluating the expression, we get: 1-9-8-7-6-5-4-3-2-1-0 = -44.

Therefore, the value of G is -44.