91Ó°ÊÓ

When dealing with fractions in algebraic equations, it's essential to know how to manipulate them. Fractions represent parts of a whole and appear frequently in algebra. In this problem, the equation involves two fractions: \(\frac{1}{4}\) and \(-\frac{5}{6}\). To solve or verify the solution of an equation containing fractions, we often must create common denominators.
This allows us to add or subtract them easily. Common denominators transform fractions into like terms, making it simple to carry out operations.
You can find a common denominator by identifying the least common multiple (LCM) of the denominators.

• In our problem, the LCM of 4 and 6 is 12.
• Convert \(\frac{1}{4}\) to \(\frac{3}{12}\) and \(-\frac{5}{6}\) to \(-\frac{10}{12}\).

Once converted, you can perform addition or subtraction straightforwardly.

Verifying a solution involves checking if the proposed solution satisfies the original equation. After substituting the value of \(x = -\frac{5}{6}\) into the equation, we simplify the expression.
We combined the fractions to calculate the left-hand side: \(\frac{3}{12} - \frac{10}{12} = -\frac{7}{12}\).
Next, compare this with the right-hand side of the equation, which is also \(-\frac{7}{12}\).

• If both sides match, your solution is correct.
• If they don't, the value substituted is not a solution.

This step is crucial in ensuring that the solution fulfills the equation's requirements.

Simplification is the process of making an equation easier to understand or solve.
It often involves reducing fractions and performing basic arithmetic operations.
In our exercise, we simplified the left side by converting fractions to have a common denominator.

• Reduce fractions: Convert denominators to be the same to perform operations easily.
• Perform operations: Calculate the addition or subtraction of fractions once denominators are alike.

This simplification enabled us to directly compare both sides of the equation. It's a fundamental process that helps clarify complex expressions and provides a clearer path to the solution.