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Understanding linear inequalities is essential for solving various algebra problems. A linear inequality looks just like a linear equation, such as ax + b = c, except it has an inequality sign (\textless, \textgreater, \textless=, \textgreater=) instead of an equal sign. These inequalities are used to represent a range of possible solutions, rather than one exact solution as in the case of equations.

Solving linear inequalities involves finding the value of the variable that makes the inequality true. For instance, in the exercise \(6a - 6 \leq -27\), 'a' is the variable, and the solution will include all the values of 'a' that make the inequality hold. One important rule to remember is that when you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign.

A linear inequality can be solved and graphed on a number line to visually represent the solution set. Starting from the inequality \(a \leq -\frac{7}{2}\), one would shade all numbers to the left of -\(\frac{7}{2}\) to show that 'a' can be any number less than or equal to -\(\frac{7}{2}\) in the solution set.

To tackle inequalities, certain algebraic methods come in handy. The step by step solution provided in the exercise is a classic example of the process used in algebra to isolate the variable of interest. This particular exercise begins with an additive operation, where a constant is added to both sides to eliminate a term. The purpose of such a maneuver is to simplify the inequality to see clearly what range of numbers satisfy it.

The next step featured is the division operation, which is crucial when the variable is multiplied by a coefficient. By dividing both sides of the inequality by the same non-zero number, we maintain the balance of the inequality while isolating the variable. Remember, preserving the direction of the inequality is only ensured when dividing by a positive number.

These two operations—addition and division—form the cornerstone of solving linear inequalities using algebraic methods. Practicing these techniques will allow one to handle even more complex inequalities that involve these along with other operations such as multiplication and subtraction.

The solution to an inequality is not just a single number but a set of numbers that satisfy the inequality. This solution set can be represented in several ways, such as using inequality notation, set-builder notation, or graphically on a number line. The final solution to the provided exercise \(a \leq -\frac{7}{2}\) tells us that any number less than or equal to -3.5 satisfies the inequality.

Graphically, you'd draw a number line, place a closed dot on -3.5 (since it includes -3.5), and shade all numbers to the left, indicating all the numbers that are part of the solution set. It is always helpful to plug a number from the solution set back into the original inequality to verify that it holds true. This not only confirms the solution but also helps in understanding the concept of inequality solutions deeply.

Moreover, solutions can be expressed in interval notation, which, for this inequality, would be \( (-\infty, -\frac{7}{2}] \). Understanding how to write and interpret this notation is fundamental as it provides a clear, concise way to represent an infinite set of numbers that qualify as solutions to an inequality.