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Visualizing inequalities with a number line is an effective way to comprehend the range of solutions. When you're given an inequality, such as the one in our exercise \(x \leq 8\), the number line helps you see all possible values that satisfy the inequality at a glance. To graph \(x \leq 8\), you'll draw a horizontal line which represents all of the real numbers.

Find the number 8 on this line. Since this inequality includes the number 8 (indicated by the 'less than or equal to' sign, \(\leq\)), you'll place a solid dot on 8 to signify that 8 is part of the solution set. Then, you would draw a thick line or shade the area to the left of 8 to represent all numbers smaller than 8, as they are also solutions to the inequality.

Solving a linear inequality is similar to solving a simple linear equation. However, instead of finding one solution, you are determining a range of solutions. In the provided problem, the process begins with eliminating the fraction and then isolating the variable \(x\) on one side.

It's crucial to perform the same operation on both sides of the inequality to maintain its balance. When you reach the final step and divide by 2 to find \(x \leq 8\), remember that the inequality symbol points to the smaller value, which is the set of numbers that \(x\) can be.

Algebraic manipulation involves various techniques used to modify and solve algebraic expressions and equations. In the context of our inequality problem, these techniques included multiplication, subtraction, and division. The key is to preserve the equality or inequality by applying the same operations to both sides.

One thing to keep in mind during manipulation is that if you ever need to multiply or divide by a negative number, you must flip the direction of the inequality symbol. Our exercise did not require this step, but it's essential to remember for solving other inequalities. Balancing the equation by subtracting \(x\) from both sides and adding 9 afterward are standard procedures to isolate \(x\) and find the solution to the inequality.