Solving equations with rational expressions involves fractions that have variables in the numerator, denominator, or both. Here’s how to solve them step by step:

Method

1. Find the Least Common Denominator (LCD):
   Identify the denominators in the rational expressions. Find the least common denominator (LCD) to eliminate the fractions.
2. Multiply Through by the LCD:
   Multiply every term in the equation by the LCD. This removes all fractions.
3. Simplify the Resulting Equation:
   Simplify the new equation. It should now be a polynomial or simpler equation.
4. Solve for the Variable:
   Solve the resulting equation using appropriate algebraic methods (e.g., factoring, isolating the variable).
5. Check your solutions:
   Substituting solutions back into the original equation may reveal undefined values (e.g., division by zero). Discard these.

Example

Solve the equation:

Step 1: Find the LCD

The denominators are $x+2$, $x-2$, and $x^2-4$. The LCD is:

Step 2: Multiply Through by the LCD

Multiply every term by $x^2-4$ to eliminate the fractions:

This simplifies to:

Step 3: Simplify

Expand and combine like terms:

Subtract $4$ from both sides:

Step 4: Solve for $x$

Factor the quadratic equation:

Solve for $x$:

Step 5: Check your solutions

Substitute $x=2$ and $x=-1$ into the original equation to ensure no division by zero, and that the equation is satisfied. Looking at the original equation

if you plug in $x=2$ you will get division by zero. So this is not a solution.

Plugging in $x=-1$

so $x=-1$ is a solution

Final Answer

There is a single solution,

Solve the equation involving rational expressions.

Give the equation in a math block.

Do not include text in the question.

Use dfrac instead of frac.

The answer should be two integer roots, that may have different values.