If \[ \begin{array}{c} \int_{-5}^{8} g(z) d z=4 \\ \int_{0}^{8} g(z) d z=10.9 \end{array} \] what does the following integral equal? \[ \int_{-5}^{0} g(z) d z= \]

Step 1 ：The integral of a function from a to b is the area under the curve of the function from a to b. The integral of g(z) from -5 to 8 is the area under the curve of g(z) from -5 to 8, and the integral of g(z) from 0 to 8 is the area under the curve of g(z) from 0 to 8.

Step 2 ：The integral of g(z) from -5 to 0 is the area under the curve of g(z) from -5 to 0. This area is the difference between the area under the curve of g(z) from -5 to 8 and the area under the curve of g(z) from 0 to 8.

Step 3 ：So, we can calculate the integral of g(z) from -5 to 0 by subtracting the integral of g(z) from 0 to 8 from the integral of g(z) from -5 to 8.

Step 4 ：Let's denote the integral of g(z) from -5 to 8 as \(\text{integral}_{-5}^{8}\), and the integral of g(z) from 0 to 8 as \(\text{integral}_{0}^{8}\). We are given that \(\text{integral}_{-5}^{8} = 4\) and \(\text{integral}_{0}^{8} = 10.9\).

Step 5 ：Then, the integral of g(z) from -5 to 0, denoted as \(\text{integral}_{-5}^{0}\), can be calculated as \(\text{integral}_{-5}^{0} = \text{integral}_{-5}^{8} - \text{integral}_{0}^{8} = 4 - 10.9 = -6.9\).

Step 6 ：Final Answer: The integral of g(z) from -5 to 0 is \(\boxed{-6.9}\).