Rational Equations: Videos & Practice Problems

Rational equations involve variables in the denominator, requiring careful handling to avoid undefined values. To solve, first identify restrictions by setting the denominator to zero. Next, eliminate fractions by multiplying through by the least common denominator (LCD). This transforms the equation into a linear form, allowing for straightforward solving. If the solution matches the restriction, it indicates no valid solution exists. Understanding these steps is crucial for effectively navigating rational equations and ensuring accurate results.

Introduction to Rational Equations

Rational equations are equations that include a variable in the denominator of a fraction. To solve these equations, the goal remains the same: find a value for \( x \) that satisfies the equation. However, it is crucial to recognize that the solution must not make any denominator equal to zero, as this would render the equation undefined. For instance, in the equation \( \frac{1}{x - 1} = 12 \), the denominator \( x - 1 \) cannot equal zero, leading to the restriction that \( x \neq 1 \).

To solve a rational equation, follow these steps:

1. **Determine Restrictions**: Identify values that would make any denominator zero. For example, if the denominator is \( x - 1 \), setting it to zero gives \( x = 1 \), indicating that \( x \) cannot be 1.

2. **Multiply by the Least Common Denominator (LCD)**: This step eliminates the fractions and transforms the rational equation into a linear equation. For example, in the equation \( \frac{x}{x - 1} = 76 \), the denominators are \( x - 1 \) and 6. The LCD is \( 6(x - 1) \). Multiply each term by the LCD to simplify the equation.

3. **Solve the Linear Equation**: After eliminating the fractions, you will have a linear equation. For instance, after multiplying, you might arrive at \( 6x = 7(x - 1) \). Distributing and rearranging terms leads to a solvable linear equation. In this case, you would distribute the 7 to get \( 6x = 7x - 7 \), then isolate \( x \) by moving terms around.

4. **Check the Solution Against Restrictions**: Finally, verify that the solution does not violate any restrictions identified in the first step. If the solution is \( x = 7 \), and since 7 does not equal 1, it is valid.

In summary, solving rational equations involves identifying restrictions, eliminating fractions through multiplication by the LCD, solving the resulting linear equation, and checking the solution against any restrictions to ensure it is valid.

Restrictions on Rational Equations

When solving rational equations, it's crucial to identify any restrictions on the variable to avoid undefined expressions. For instance, if you have an equation like \(\frac{x - 5}{x - 2} = \frac{-3}{x - 2} + 6\), the first step is to determine the restrictions by identifying values that make the denominator zero. In this case, setting \(x - 2 = 0\) reveals that \(x\) cannot equal 2.

Next, to eliminate the fractions, multiply every term by the least common denominator (LCD), which in this example is \(x - 2\). Distributing the LCD across the equation simplifies it significantly. After canceling the denominators, you would be left with a linear equation: \(x - 5 = -3 + 6(x - 2)\). Distributing the 6 gives you \(x - 5 = -3 + 6x - 12\), which simplifies to \(x - 5 = 6x - 15\).

To solve for \(x\), rearrange the equation by moving all \(x\) terms to one side and constants to the other. This involves adding 15 to both sides and subtracting \(x\) from both sides, resulting in \(10 = 5x\). Dividing both sides by 5 yields \(x = 2\).

However, upon checking this solution against the restriction, you find that \(x = 2\) is not valid since it violates the initial restriction. Therefore, when a solution equals a restriction, it indicates that there is no solution to the equation. In mathematical terms, this can be expressed as the solution set being the empty set, denoted as \(\emptyset\). This highlights the importance of checking solutions against restrictions to ensure they are valid.

Solve the equation. $\frac{2x+4}{x-1}=5$

• A. $x=3$
• B. $x=1$
• C. $x=5$
• D. No solution 

Answer: (A)

Solve the equation. $\frac{5}{x}-\frac{2}{3x}=4+\frac{3}{x}$

• A. $x=0$
• B. $x=1$
• C. $x=\frac13$
• D. No solution 

Answer: (C)

Solve the equation. $\frac{-5}{x+4}-3=\frac{x-1}{x+4}$

• A. $x=4$
• B. $x=1$
• C. $x=-4$
• D. No solution

Answer: (D)

Here’s what students ask on this topic:

What is a rational equation and how do you solve it?

A rational equation is an equation that contains one or more fractions with variables in the denominator. To solve a rational equation, follow these steps:

1. Identify restrictions by setting the denominator equal to zero and solving for the variable. These are values that the variable cannot take.
2. Multiply both sides of the equation by the least common denominator (LCD) to eliminate the fractions.
3. Simplify the resulting equation, which should now be a linear equation.
4. Solve the linear equation for the variable.
5. Check the solution against the restrictions to ensure it is valid.

If the solution matches any of the restrictions, then there is no valid solution.

How do you find the least common denominator (LCD) in a rational equation?

To find the least common denominator (LCD) in a rational equation, follow these steps:

1. Identify all the denominators in the equation.
2. Factor each denominator into its prime factors.
3. For each unique factor, take the highest power that appears in any of the denominators.
4. Multiply these highest powers together to get the LCD.

The LCD is the smallest expression that all denominators can divide into without leaving a remainder. Multiplying through by the LCD will eliminate the fractions in the equation.

What are restrictions in rational equations and why are they important?

Restrictions in rational equations are values that the variable cannot take because they make the denominator zero, leading to undefined expressions. To find restrictions, set each denominator equal to zero and solve for the variable. These values are important because they ensure the equation remains valid and defined. When solving rational equations, always check that your solutions do not match any of the restrictions. If a solution matches a restriction, it means there is no valid solution for the equation.

What should you do if your solution to a rational equation matches a restriction?

If your solution to a rational equation matches a restriction, it means that the solution is not valid because it makes the denominator zero, leading to an undefined expression. In such cases, the equation has no solution. You can denote this by stating that the solution set is empty or by writing 'no solution.' Always check your solutions against the restrictions to ensure they are valid.

Can you provide an example of solving a rational equation step-by-step?

Sure! Let's solve the rational equation $\frac{x}{\mathrm{x-1}}=\; 7/6$:

1. Identify restrictions: Set the denominator $\mathrm{x-1}$ to zero. $\mathrm{x-1=0}$ gives $\mathrm{x=1}$. So, $x$ cannot be 1.
2. Multiply by the LCD: The LCD is $\mathrm{6(x-1)}$. Multiply both sides by the LCD to eliminate fractions.
3. Simplify: $\frac{\mathrm{6(x-1)x}}{\mathrm{x-1}}=\; 7$ simplifies to $\mathrm{6x=7}$.
4. Solve the linear equation: $\mathrm{x=7/6}$.
5. Check the solution: $\mathrm{7/6}$ is not equal to the restriction $1$, so it is valid.

The solution is $\mathrm{x=7/6}$.