When is it appropriate in a math problem to use the "almost equal to" sign as opposed to the regular one?

Answer 1

Most commonly when taking something that is expressed as a radical or other irrational number and you turn it into a decimal approximation.

For example:
2 + x² = 5
x² = 3
x ≈± 1.732

Problems involving pi, for example often need you to use the "almost equal to sign"

Radius = 9 ft.
Area of the circle = 81π ≈ 254.47 sq. ft.

Or if you do a lot of computations and get to a long value and need to reduce it to significant digits, you might do that too.

5 x 10^7 * 3.521827 + 4.4123123 ≈ 180 million