Question

solve for x in a given interval using guess improvement stategy. express to the nearest tenths. cos x=1/x, -4<=x<=0

Answer 1

there are lots of different iterative solution methods:

bisection
Newton's method,
secants,
etc.

give us a hint where you want to go.

Answer 2

for these kind of problems, it is often a good idea to get a ball-park answer by graphing both the left side and the right side of the equation in the given domain.
A good webpage for this is
http://rechneronline.de/function-graphs/

In the first graph box enter
cos(x)
in the second graph box enter
1/x
It will show that there is an answer near -2 and -4.5
the latter being outside the domain.

You can then fine-tune the solution by zooming in.
After a few intermediate changes, I changed
"range x-axis from" -2.08 to -2.06
"range y-axis from" -.5 to -.45
to get x = appr. -2.074

check:
cos(-2.074) = -.48223
1/-2.074 = -.48216
not bad