Exercises: Solve Rational and Radical Equations

Which value of $x$ must be excluded from the domain of the equation $\frac{3}{x - 4} = 7$?

What is the domain restriction for the equation $\sqrt{2x - 6} = 4$?

A student solves an equation and arrives at $x = 5$. When they substitute $x = 5$ into the original equation, they get $\frac{5}{0} = \frac{3}{0}$. What does this tell them?

Solve the rational equation $\frac{6}{x} = 2$.

Enter the value of $x$. (Note: identify and exclude any domain restrictions in your work before solving.)

Solve $\frac{x}{x - 5} = \frac{3}{x - 5}$.

Enter the value of $x$ that is the solution (or enter 0 if there is no solution).

Solve the rational equation $\frac{2}{x - 3} = \frac{8}{x - 3}$. Show your domain restriction, your algebraic steps, and explain the result.

Solve $\sqrt{3x - 2} = 4$. Show all steps: state the domain, isolate the radical (if needed), square both sides, solve, and check.

Solve $\sqrt{x + 6} = x$. Show the domain restriction, all algebraic steps, and check all solutions. Identify any extraneous solutions.