Rational Equations: Videos & Practice Problems

Which of the following is a rational equation?

Which of the following statements correctly distinguishes a rational equation from other equations?

Find the restrictions, if any, on $x$ for the equation $\frac{x-2}{x^2-5x+6}=\frac{1}{x-3}$.

Solve the following rational equation for $x$:

$\frac{1}{x^2-5x+6}+\frac{1}{x-2}=\frac{1}{x+3}$

For the equation $\frac{1}{x+1}+\frac{2}{x-2}=\frac{3}{x^2-x-2}$, what linear equation results after clearing the denominators?

Solve the rational equation $\frac{2}{x}+\frac{3}{x+2}=1$:

You solved $\frac{x+1}{x-2}=3$ and found $x=5$. Verify by substituting $x=5$ back into the original equation and state whether it is an acceptable solution considering any restrictions.

Solve $\frac{x-3}{x-2}=\frac{5}{x-2}$.

Solve $\frac{3}{x+1}-\frac{2}{x^2-1}=\frac{1}{2}$.

Which justification best explains why multiplying both sides of a rational equation by the LCD is a valid step when solving rational equations (provided you track restrictions)?

Solve the rational equation $\frac{x+1}{x-2}+\frac{2}{x+1}=\frac{4}{x-2}$.

Identify domain restrictions, if any, for the rational expression $\frac{5}{x^2+x-12}$.

Solve for $x$:

$\frac{5}{x^2-9}+\frac{1}{6}=\frac{1}{x-3}$

Multiply through by the LCD and simplify to the linear equation for $\frac{1}{x+2}-\frac{3}{x+1}=\frac{2}{x^2+3x+2}$. What linear equation do you obtain?

Which sentence best defines an extraneous solution in the context of rational equations?

Solve the rational equation $\frac{3x}{x^2-x-6}=\frac{2}{x+2}$.

After solving a rational equation you obtain $x=-1$ as a candidate root. The original equation has factors of $(x+1)$ in some denominators. What conclusion should you reach and why?

Solve the rational equation $\frac{1}{6}+\frac{1}{x}=\frac{1}{4}$ for $x$.

What is the simplest correct description of the least common denominator (LCD) used when solving rational equations?

Solve for $x$:

$\frac{x+2}{x^3-x^2-6x}+\frac{1}{x}=\frac{1}{x-3}$

Which of the following sequences is the correct and complete order of actions to solve a nontrivial rational equation with variable denominators?

If clearing denominators leads to an identity like $0=0$ but the original equation had restrictions $x\ne 2$ and $x\ne -1$, what is the correct description of the solution set?

Solve $\frac{x^2-x-2}{x-2}=5$ for $x$: