Using linearity and bounds-reversal to evaluate a combined integral

Since the integral runs from $5$ to $1$ instead of $1$ to $5$, we use the reversal property $\int_5^1 h(x)\,dx = -\int_1^5 h(x)\,dx$. So $\int_5^1 (2f(x)-3g(x))\,dx = -\int_1^5 (2f(x)-3g(x))\,dx.$

Now apply linearity to the integral on $[1,5]$: $\int_1^5 (2f(x)-3g(x))\,dx = 2\int_1^5 f(x)\,dx - 3\int_1^5 g(x)\,dx = 2(8) - 3(3) = 16 - 9 = 7$. Negating this (from the bounds reversal) gives the final answer $-7$.