Identifying the domain of a variable in an equation

The domain is the collection of values a variable may take without breaking the equation.
Most of the time, every value is allowed except a few special ones.
Your job is to find and exclude those.

The most common restriction is division by zero.
Any value that makes a denominator equal to zero is not allowed.
In $\frac{5}{x - 2}$, the value $x = 2$ is excluded, so the domain is all values except $2$.

A second restriction comes from square roots.
The expression under a square root cannot be negative, so it must be zero or more.
In $\sqrt{x - 4}$, you need $x - 4 \ge 0$, which gives $x \ge 4$.

Word problems add their own limits — a count of people or boxes must be a whole number that is zero or more.
And after solving, check each answer in the original equation and throw out any that do not actually work.
For example, if $\frac{y}{\sqrt{y + 2}}$ must be positive, a negative value of $y$ is not valid.

Which value is not allowed for $x$ in $\frac{7}{x + 1}$?
The denominator $x + 1$ cannot be zero, so $x$ cannot be $-1$.
The domain is all values except $x = -1$.

What is the domain of $\sqrt{2x - 6}$?
The inside must be zero or more: $2x - 6 \ge 0$, so $2x \ge 6$ and $x \ge 3$.
The domain is all values with $x \ge 3$.

Solving $\frac{y}{\sqrt{y + 2}} = \frac{7}{3}$ gives $y = 7$ or $y = -\frac{14}{9}$. Which is valid?
The right side is positive, so the left side must be positive too, which needs $y > 0$.
So $y = -\frac{14}{9}$ is rejected, and only $y = 7$ is valid.