Solving Rational Equations: A Step-by-Step Guide

Rational equations are equations that involve fractions with variables in the denominator. Solving them can seem daunting at first, but with the right approach, it becomes a manageable process. This guide will walk you through the steps involved in solving rational equations, using clear examples and explanations.

Understanding Rational Equations

A rational equation is an equation where at least one term contains a variable in the denominator of a fraction. Here's an example:

```
1/(x + 2) + 3/(x - 1) = 2
```

In this equation, the variables 'x' appear in the denominators of the fractions.

Steps to Solve Rational Equations

Here's a step-by-step guide to solving rational equations:

1. Find the Least Common Multiple (LCM)

The first step is to find the LCM of all the denominators in the equation. This will help us eliminate the fractions.

For example, in the equation above, the LCM of (x + 2) and (x - 1) is simply their product: (x + 2)(x - 1).

2. Multiply Both Sides by the LCM

Multiply both sides of the equation by the LCM you found in step 1.

In our example, we would multiply both sides by (x + 2)(x - 1):

```
(x + 2)(x - 1) * [1/(x + 2) + 3/(x - 1)] = 2 * (x + 2)(x - 1)
```

3. Simplify the Equation

Simplify the equation by canceling out the common factors in the numerator and denominator. This will leave you with a polynomial equation.

In our example, after simplification, we get:

```
(x - 1) + 3(x + 2) = 2(x + 2)(x - 1)
```

4. Solve the Polynomial Equation

Now you have a polynomial equation. Solve it by simplifying, combining like terms, and using techniques like factoring or the quadratic formula.

Continuing with our example, we would expand the equation, combine terms, and set it equal to zero to get a quadratic equation. Then, we could solve it using the quadratic formula.

5. Check for Extraneous Solutions

It's important to check your solutions by plugging them back into the original equation. This is because some solutions might make the denominator of the original equation equal to zero, which is undefined. These solutions are called extraneous solutions and must be discarded.

Examples

Example 1:

Solve the equation:

```
2/(x - 1) + 1/(x + 1) = 3
```

1. LCM: (x - 1)(x + 1)

2. Multiply both sides by the LCM:

```
(x - 1)(x + 1) * [2/(x - 1) + 1/(x + 1)] = 3 * (x - 1)(x + 1)
```

3. Simplify:

```
2(x + 1) + (x - 1) = 3(x - 1)(x + 1)
```

4. Solve the polynomial equation:

```
2x + 2 + x - 1 = 3(x^2 - 1)
3x + 1 = 3x^2 - 3
3x^2 - 3x - 4 = 0
```

Solving the quadratic equation, we get x = (3 ± √57)/6.

5. Check for extraneous solutions: Both solutions are valid.

Example 2:

Solve the equation:

```
1/x + 2/(x + 3) = 1
```

1. LCM: x(x + 3)

2. Multiply both sides by the LCM:

```
x(x + 3) * [1/x + 2/(x + 3)] = 1 * x(x + 3)
```

3. Simplify:

```
(x + 3) + 2x = x(x + 3)
```

4. Solve the polynomial equation:

```
3x + 3 = x^2 + 3x
x^2 = 3
x = ±√3
```

5. Check for extraneous solutions: Both solutions are valid.

Applications

Rational equations have numerous applications in various fields, including:

Physics: Solving for variables in equations describing motion, electricity, and other physical phenomena.
Chemistry: Calculating concentrations and reaction rates.
Economics: Modeling supply and demand curves.
Engineering: Designing structures and systems.

Conclusion

Solving rational equations involves a systematic approach that involves finding the LCM, simplifying the equation, and checking for extraneous solutions. Mastering this skill is essential for success in various fields of study and practical applications.

Solving Rational Equations: A Step-by-Step Guide