Question

show that the equation | x-1 | + cos (x) = 3
has at least two solutions using Interneduate-value theorem

Answer 1

let f(x) = |x-1| + cos x - 3

Then the equation you've written down has solutions if f(x) = 0

Now f(x) is continuous and to use the IVP, we want to find a couple of pairs of points, a and b, such that
f(a) < 0 and f(b) > 0
then it means for x that between a and b, f(x) = 0

For example,

f(2pi) = 2pi - 1 + 1 - 3 = 2pi - 3 > 0
and
f(0) = 1 + 1 - 3 = -1 < 0

Hence between 0 and 2pi, f(x) must have a zero

Now find another pair x = 0 and x = some positive number