1. Equations and Inequalities

Rational Equations

1. Equations and Inequalities

Rational Equations: Videos & Practice Problems

Topic summary

Rational equations involve variables in the denominator, requiring careful handling to avoid undefined values. To solve, first identify restrictions by setting denominators to zero. Next, eliminate fractions using the least common denominator (LCD), transforming the equation into a linear form. After solving the linear equation, check if the solution violates any restrictions. If it does, the equation has no solution. Understanding these steps is crucial for effectively solving rational equations and ensuring valid solutions.

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Introduction to Rational Equations

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Introduction to Rational Equations Video Summary

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. A key aspect of rational equations is that the solution must not make any denominator equal to zero, as this would render the equation undefined. For example, in the equation $ \frac{1}{x - 1} = 12 $, the denominator $ x - 1 $ cannot equal zero, which leads to the restriction that $ x \neq 1 $.

To solve a rational equation, follow these steps:

1. **Determine Restrictions**: Identify values that would make the denominator zero. For instance, if the equation is $ \frac{x}{x - 1} = 76 $, set the denominator $ x - 1 = 0 $ to find 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 the example above, the denominators are $ x - 1 $ and 6, and since they have no common multiples, the LCD is $ 6(x - 1) $. Multiply every term in the equation by this LCD to simplify.

3. **Solve the Linear Equation**: After eliminating the fractions, you will have a linear equation. For example, after multiplying, you might simplify to $ 6x = 7(x - 1) $. Distributing the 7 gives $ 6x = 7x - 7 $. Rearranging the equation leads to $ -x = -7 $, which simplifies to $ x = 7 $.

4. **Check the Solution Against Restrictions**: Finally, verify that the solution does not violate any restrictions. In this case, since $ x = 7 $ does not equal 1, the solution 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. This systematic approach allows for effective resolution of rational equations.

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Restrictions on Rational Equations

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Restrictions on Rational Equations Video Summary

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 are left with a linear equation: $x - 5 = -3 + 6(x - 2)$. Distributing the 6 gives $x - 5 = -3 + 6x - 12$, 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 permissible since it was identified as a restriction. This indicates that the equation has no solution. In mathematical terms, the solution set is the empty set, denoted as $\emptyset$, meaning there are no valid values for $x$ that satisfy the original equation.

In summary, when a solution coincides with a restriction, it signifies that the rational equation has no solution, reinforcing the importance of checking solutions against identified restrictions.

Problem

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

$x=3$

$x=1$

$x=5$

No solution

Problem

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

$x=0$

$x=1$

$x=\frac13$

No solution

Problem

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

$x=4$

$x=1$

$x=-4$

No solution

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Rational Equations definitions
1. Equations and Inequalities
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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:

Identify restrictions by setting each denominator equal to zero and solving for the variable. These are values that the variable cannot take.
Find the least common denominator (LCD) of all the fractions in the equation.
Multiply every term in the equation by the LCD to eliminate the fractions.
Solve the resulting linear equation.
Check the solution against the restrictions. If the solution violates any restriction, the equation has no solution.

How do you determine the restrictions in a rational equation?

To determine the restrictions in a rational equation, set each denominator equal to zero and solve for the variable. These values are the restrictions because they make the denominator zero, which would make the fraction undefined. For example, in the equation $\frac{x}{x - 1} = 12$ , the restriction is found by setting $x - 1 = 0$ , giving $x = 1$ . Therefore, $x$ cannot be 1.

What is the least common denominator (LCD) and how is it used in solving rational equations?

The least common denominator (LCD) is the smallest multiple that is common to all denominators in a rational equation. It is used to eliminate the fractions by multiplying every term in the equation by the LCD. This transforms the rational equation into a linear equation, which is easier to solve. For example, in the equation $\frac{x}{x - 1} = 12$ , the LCD is $x - 1$ . Multiplying every term by $x - 1$ eliminates the fraction.

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

If your solution to a rational equation equals the restriction, then the equation has no solution. This is because the restriction represents a value that makes the denominator zero, which is undefined. For example, if you solve the equation and find $x = 2$ , but the restriction is $x \neq 2$ , then there is no valid solution. In such cases, you can state that the solution set is empty or that there is no solution.