91Ó°ÊÓ

Algebraic equations are mathematical statements that show the equality of two expressions.
They often contain variables which represent unknown values.
In our example, the variable is \( x \).
When dealing with algebraic equations, especially those involving fractions, it's crucial to understand the format and operations involved.
Our equation is \[ \frac{5}{6x} - \frac{8}{2x} = \frac{x}{4} \].
The goal is to solve for \( x \) by finding values that satisfy the equality.
This sometimes involves manipulating the equation through addition, subtraction, multiplication, or division to isolate the variable. Adjustments need to be done carefully to maintain the equation's balance.

In any mathematical function, the domain refers to all possible input values (typically represented by the variable \( x \)) that the function can accept without leading to undefined results.
Conversely, the range represents all possible output values resulting from plugging in the domain values.
For our equation, we need to find the values of \( x \) that don't make the equation undefined.
This is done by focusing on the denominators of fractional parts. Any value that makes a denominator zero is excluded from the domain.
For instance, in our example equation \[ \frac{5}{6x} - \frac{8}{2x} = \frac{x}{4} \], we identify denominators \( 6x \) and \( 2x \) and ensure they are not zero.
By setting them not equal to zero, \( 6x eq 0 \) and \( 2x eq 0 \) we find \( x eq 0 \). Hence, the domain of \( x \) is all real numbers except zero.

Undefined fractions occur when the denominator is zero.
This is because division by zero is undefined in mathematics.
In our equation, two terms have denominators involving \( x \): \[ \frac{5}{6x} \text{ and } \frac{8}{2x} \].
To find the domain of the variable \( x \), we need to ensure that neither denominator equals zero.
Setting up the conditions \( 6x eq 0 \) and \( 2x eq 0 \), we solve these inequalities and find \( x eq 0 \).
Therefore, \( x \) cannot be zero because it would make the equation undefined.
As a result, the domain of \( x \) excludes zero, ensuring all fractions are properly defined.