91Ó°ÊÓ

When solving equations, especially those with negative exponents, the goal is to find the value of the variable that makes the equation true. Let's take a closer look at this process. In the given equation, the variable terms are intertwined with negative exponents, such as \((x+1)^{-1}\). To deal with this form, it's best to eliminate the negative exponent first. This helps to simplify the expression and makes solving easier.

To remove the negative exponent, multiply every term by the base of the negative exponent—in this case, \((x+1)\). Doing this turns the equation into a linear format, like in the given example:

• Original: \(3+x(x+1)^{-1}=2(x+1)^{-1}\)
• After multiplying through by \((x+1)\): \(3(x+1) + x = 2\)

This crucial step transitions an otherwise complex expression into a more straightforward algebraic problem.

Simplifying expressions involves combining like terms and expanding where necessary. Once negative exponents are cleared, as shown in the previous section, the expression becomes easier to handle. By expanding the term \(3(x+1)\), we simplify the expression further:

\[3(x+1) = 3x + 3\]

This turns the expression into \(3x + 3 + x\). Simplifying this expression requires you to combine like terms, which are terms with the same variable raised to the same power. Combining \(3x\) and \(x\) gives us \(4x\), so the equation simplifies to:

\[4x + 3 = 2\]

When expressions are simplified, solving the equation becomes straightforward because the relationship between the variable and constants is clearer.

Algebraic manipulation is the process of strategically moving terms around in an equation to isolate the variable you are solving for. Once the expression is simplified, as shown previously, the next step is to isolate \(x\). Methods like adding, subtracting, multiplying, or dividing both sides of the equation are key.

Starting from \(4x + 3 = 2\), we subtract 3 from both sides to get:

\[4x = 2 - 3\]

This simplifies to \(4x = -1\). The final step involves dividing both sides by 4 to solve for \(x\):

\[x = \frac{-1}{4}\]

Using such straightforward manipulation, we can solve for \(x\) and successfully complete the equation. Carefully tracking these steps ensures accuracy, and improving mastery over these basic algebraic operations is crucial for tackling more complex problems later on.