Root with bolzano theorem

Question

Given this equation $a\cos{x}+b=x$ with $a,b>0$ how to prove that there is at least one root between $(0,a+b]$ ?
For $x=0$ its $a+b$ which is >0
For $x=a+b$ its $a\cos(a+b) -a=a(\cos(a+b)-1)\leq0$
The problem here is that it is less than equal not less equal. Any idea what should I do ?