Let $$f(x) =2x-5$$ and $$g(x) =7-2x$$. Then |f(x)+ g(x)| = |f(x)|+ |g(x)| if and only if

$$|f(x)+ g(x)| = |f(x)| + |g(x)|$$ if and only if

case 1: $$f(x) \geq 0$$ and $$g(x) \geq 0$$
<=> $$ 2x-5 \geq 0 $$ and $$7-2x \geq 0$$
<=> $$ x \geq \frac{5}{2}$$ and $$ \frac{7}{2} \geq x$$ 

<=> $$\frac{5}{2}\leq x\leq\frac{7}{2}$$

case 2: $$f(x) \leq 0$$ and $$g(x) \leq 0$$

<=> $$ 2x-5 \leq 0 $$ and $$7-2x \leq 0$$
<=> $$ x \leq \frac{5}{2}$$ and $$ \frac{7}{2} \leq x$$
 So x<=5/2 and x>=7/2 which is not possible.

Hence, answer is 

<=> $$\frac{5}{2}\leq x\leq\frac{7}{2}$$