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Understanding how to isolate the variable is a fundamental skill in solving inequalities. It involves manipulating an inequality equation so that the variable we are solving for is on one side, and all other terms are on the other side.

Let's break it down with our example, \( \frac{1}{5} x>3 \). Here, our goal is to isolate \(x\). Since \(x\) is multiplied by \( \frac{1}{5} \), we do the reverse operation: we multiply both sides of the inequality by 5. This helps eliminate the fraction and makes \(x\) the subject of the equation. The inequality now reads \(x > 15\). Remember, when we multiply or divide both sides of an inequality by a positive number, the direction of the inequality remains the same. However, if we were to multiply or divide by a negative number, we would need to reverse the inequality symbol.

Inequality graphing is a visual way to represent solutions to an inequality on a number line. This is helpful because it allows you to see at a glance all the possible values that solve the inequality.

In the given example, we have already isolated the variable, and our inequality is \(x>15\). To graph this, we start by drawing a number line, then we locate the number 15 on this line. Since our inequality does not include 15 (it's strictly greater than, not greater than or equal to), we'll use an open circle to mark this value. An open circle means that the actual number 15 is not a solution.

From there, we draw an arrow extending to the right from the open circle. This arrow indicates that all numbers to the right of 15 are part of the solution set, so every number greater than 15 satisfies the inequality.

Number line representation is a straightforward way to display the set of solutions for inequalities. It illustrates clearly which numbers are included and which are not, with simple symbols like open and closed circles.

The key in representing an inequality on a number line is to first determine if the inequality is 'strict' (>, <) or 'non-strict' (\geq, \leq). For strict inequalities, where the variable does not include the value it's being compared to (as in our example, \(x>15\)), we use an open circle on the number line.

However, if the inequality was non-strict (for example, \(x \geq 15\)), we would use a closed circle to include 15 as a solution. Always remember, after placing the correct circle, draw an arrow in the direction where the inequality holds. This visual cue immediately lets you know the range of the solution set.