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Understanding algebra often starts with the basic skill of translating word problems into mathematical expressions. This process involves identifying keywords and phrases in a problem statement, and representing them using algebraic symbols.

For example, in the exercise 'Six less than 3 times a number is 9', the phrase 'six less than' indicates a subtraction, and '3 times a number' suggests multiplication. We represent 'a number' with a variable, typically 'x' or 'y'. Thus, we can translate the statement into the equation: \(3x - 6 = 9\).

It is critical to comprehend the order of operations implied by these phrases. For instance, 'six less than' comes after the multiplication, hence the subtraction is performed after multiplying the number by three. By regularly practicing this translation, students can better approach and solve algebraic problems.

Integer operations are the mathematical operations of addition, subtraction, multiplication, and division involving integers. Integers include all whole numbers and their negatives, as well as zero.

When solving algebraic equations involving integers, it's crucial to follow the rules for integer operations. For instance, when we add the opposite of a negative number, we actually perform addition. Similarly, subtracting a negative is equivalent to adding a positive. In our problem, we use addition to 'undo' the subtraction of six to isolate the variable on one side of the equation: \(3x - 6 + 6 = 9 + 6\), thus simplifying to \(3x = 15\).

Understanding how to manipulate integers within equations is an essential aspect of algebra that allows students to simplify and solve for the unknown.

Isolating the variable is a cornerstone practice in solving algebraic equations, which involves getting the unknown variable on one side of the equation and all other numeric values on the opposite side.

In our exercise, after simplifying to \(3x = 15\), the next step is to isolate 'x' by performing an operation that will leave it by itself. Since 'x' is being multiplied by three, we perform the inverse operation, which is division, to both sides of the equation. By dividing both sides by the number 3, we are applying the principle of equivalence, which states that if we do the same thing to both sides of an equation, it still holds true:

\[ x = \frac{15}{3} \]

After the calculation, we find that \(x = 5\), which is the solution to our equation. Mastery of isolating variables is a foundational skill that allows students to progress to more complex algebraic concepts and equations.