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Cross-multiplication is a handy technique in solving rational equations that involve fractions. It enables us to clear the fractions, turning the equation into a more straightforward linear form. When you have a rational equation like the one in our exercise:

• \(\frac{6}{x+3}=\frac{4}{x-3}\)

the goal is to eliminate the denominators. Cross-multiplication works by multiplying the numerator of each fraction by the denominator of the other fraction, across the equal sign. Here, the process gives us:

• \(6*(x-3)=4*(x+3)\)

By doing this, we're effectively saying that since the fractions are equal, their cross-products must also be equal. This results in a simplified equation without fractions, which is much easier to manage. With the fractions cleared, we are now ready to tackle further simplification using the next concept: the distributive property.
Always ensure both sides of the equation are fully multiplied out before proceeding. This sets a solid foundation for solving rational equations.

The distributive property is essential when you see an expression in the form of a product that involves sums or differences inside parentheses. After cross-multiplying, we have:

• \(6*(x-3)\) and \(4*(x+3)\)

Using the distributive property, you multiply each term inside the parentheses by the number outside. Here's how it unfolds for our equation:

• \(6x - 18\) (from \(6*(x-3)\))
• \(4x + 12\) (from \(4*(x+3)\))

Once these expressions are simplified, our equation becomes \(6x - 18 = 4x + 12\).
This step is crucial as it breaks down complex expressions into simple, manageable terms. Using the distributive property ensures no part of the expressions is overlooked, which propels us to the next stage of isolating the variable.
Remember, be systematic when applying distribution. Keep track of signs – they can change the entire meaning of the equation!

Isolating the variable is a key phase in solving any equation, involving getting the unknown variable (in this case, \(x\)) by itself on one side of the equation. After using the distributive property, our equation is:

• \(6x - 18 = 4x + 12\)

The objective is to collect all \(x\)-terms on one side and constants on the other. Here’s how you do it:

• Subtract \(4x\) from both sides: \(6x - 4x - 18 = 12\)
• This simplifies to: \(2x - 18 = 12\)

Next, eliminate the constant from the left side by adding 18 to both sides, resulting in:

• \(2x = 30\)

Finally, solve for \(x\) by dividing both sides by 2:

• \(x = 15\)

This process of isolating the variable enables us to arrive at the answer efficiently. By breaking down each step, it becomes clearer how to manipulate the equation until \(x\) stands alone. This approach, along with careful arithmetic, ensures accuracy in finding the solution to rational equations. Always double-check each maneuver to avoid mistakes.