Question

If \(2+\frac{2}{x}=3-\frac{3}{x}\) , then \(x=\) A $$-5$$ B $$\frac{2}{3}$$ C $$1$$ D. $$\frac{3}{2}$$ E $$5$$

Short answer

x = 5

Step-by-step solution

Isolate the fraction terms

To begin solving the equation, move all the fraction terms to one side of the equation. This gives us: \[ 2 + \frac{2}{x} + \frac{3}{x} = 3 \]

Simplify the equation

Combine the fraction terms: \[ 2 + \frac{2 + 3}{x} = 3 \ 2 + \frac{5}{x} = 3 \]

Isolate the variable term

Subtract 2 from both sides of the equation to isolate the term with the variable: \[ \frac{5}{x} = 1 \]

Solve for x

Multiply both sides of the equation by x to solve for x: \[ 5 = x \ x = 5 \]

Key concepts

Fraction Isolation

To solve the initial equation, we first need to isolate the fractions. This means rearranging the equation to get all the fraction terms on one side. In the problem \(2 + \frac{2}{x} = 3 - \frac{3}{x}\), we move all fraction-related terms to one side. This step helps in simplifying the problem as we work towards isolating the variable.
Here's how we do it step by step:
Move \( \frac{3}{x} \) from one side to add it to the fraction \( \frac{2}{x}\):
\[ 2 + \frac{2}{x} + \frac{3}{x} = 3 \]
This step is fundamental because having all the fractions on one side allows us to manage them more easily in subsequent steps.

Variable Isolation

Once we have isolated the fractions, our next task is to isolate the variable, which is 'x' in this case. We start by combining fraction terms:
\[ 2 + \frac{2 + 3}{x} = 3 \]This simplifies our equation to:
\[ 2 + \frac{5}{x} = 3 \]
The key here is to get the term \( \frac{5}{x}\) by itself. To do this, we need to move the other constants out of the way by performing basic algebraic manipulations. Subtract 2 from both sides:
\[ \frac{5}{x} = 1 \]
At this point, the variable \(x\) is neatly isolated within the fraction, preparing us for the final solving step.

Solving Equations

Having isolated the variable, our final step is to solve for it directly. From our modified equation:
\[ \frac{5}{x} = 1 \]
We need to eliminate the fraction by getting rid of the denominator. To do this, multiply both sides of the equation by 'x':
\[ 5 = x \]
This tells us directly that \(x = 5\). This simple multiplication step is crucial in not only solving the variable but also ensuring the fraction is removed correctly.
Finally, verifying by substituting back into the original problem is a good practice. Plugging \(x = 5\) into the original equation confirms that both sides equal the same value.