continuous function?

Question

f(x) is a continuous function.
for all x |f(x)|<(or equals) 7
show that f(x)+2x=3 has at least one solution

could someone please explain this to me
show your work
thanks!

Answer 1

-7<= f(x) <= 7

-10+2x<= f(x)+2x-3 <= 4+2x
you can sea that for x= -4 f(x) +2x-3 <0 and for x= 6 this expression is positive
So as f(x)+2x-3 is continuous it must take at least once the value 0