91Ó°ÊÓ

When solving rational equations, the first crucial step is to clear the denominators. This process involves eliminating the fractions by finding a common denominator and multiplying every term by it.

By doing so, we convert the equation into a simpler form without fractions, making subsequent steps more manageable. In the given example, the common denominator is \(x+1\). By multiplying both sides of the equation \( \frac{3x}{x+1} - 5 = \frac{x-11}{x+1} \) by \( x+1 \), we eliminate the denominators:

\[ (x+1) \left( \frac{3x}{x+1} - 5 \right) = (x+1) \frac{x-11}{x+1} \]
This simplifies to:

\[ 3x - 5(x+1) = x - 11 \]
This process is known as 'clearing the denominators' and sets the stage for easier algebraic manipulation in the following steps.

After clearing denominators, the next step is to isolate the variable. This involves rearranging the equation so that the variable 'x' is on one side and the constants are on the other.

Start by simplifying the equation obtained after clearing denominators:

\[ 3x - 5(x + 1) = x - 11 \]
Expand and combine like terms:

\[ 3x - 5x - 5 = x - 11 \]
This further simplifies to:

\[ -2x - 5 = x - 11 \]
Next, move all terms with 'x' to one side:
\[ -2x - x = -11 + 5 \]
This simplifies further to:

\[ -3x = -6 \]
Finally, divide by -3 to isolate x:
\[ x = 2 \]
Isolating variables is a vital part of solving equations, allowing us to find the value of the unknown.

Classifying equations helps in understanding the nature of their solutions. There are three main types:

• Identity
• Inconsistent
• Conditional

An identity equation is true for all values of the variable. An inconsistent equation has no solution as it's never true. A conditional equation, like the one in this exercise, is only true for specific values of the variable.

For the given equation, after solving we found \( x = 2 \). To confirm, substitute \( x = 2 \) back into the original equation:

\[ \frac{3(2)}{2+1} - 5 = \frac{2-11}{2+1} \]
Simplify each side:
\[ \frac{6}{3} - 5 = \frac{-9}{3} \]
This results in:
\[ 2 - 5 = -3 \]
Both sides are equal, confirming \( x = 2 \) is the correct solution, thus it's a conditional equation.

Understanding how to classify equations can help in determining the types of solutions to expect.