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Graphing inequalities is a useful way to visualize the set of possible solutions. When you solve an inequality like \( -3x - 1 > 14 \), you're finding the values that \( x \) can take to make the inequality true. After solving the problem step by step and finding \( x < -5 \), it's time to represent this information on a graph. To do this, draw a horizontal line, which will be our number line. Place a point at \( -5 \) on this line.

Since \( -5 \) is not part of the solution (as indicated by the '<' sign rather than '\(\backslash\backslashle\)'), the point is not filled in but rather left as an open dot. From this open dot, draw a line to the left, showing that all numbers to the left of \( -5 \) (which are smaller than \( -5 \)) are included in the solution set. This simple visual representation can provide an immediate understanding of the possible solutions to an inequality.

When dealing with inequalities, dividing by a negative number has a crucial distinction from dividing by a positive number. Normally, when you divide both sides of an equation by the same positive number, the direction of the inequality remains unchanged. However, if you divide both sides of an inequality by a negative number, you must flip the inequality sign for the inequality to remain valid.

This is because of the rules governing the order of numbers on a number line; for example, if \( a < b \), then after multiplying or dividing both \( a \) and \( b \) by a negative number, \( -a \) will be greater than \( -b \) because negative multiplication or division reverses the order. So, in the given exercise, when you divide \( -3x > 15 \) by \( -3 \), it becomes \( x < -5 \) instead of \( x > -5 \) because the sign flips.

The number line is a fundamental tool for understanding and representing mathematical concepts. In our case, it helps to represent solutions to inequalities. On this line, every point corresponds to a real number, and the position of the number indicates its value relative to other numbers. For the inequality \( x < -5 \), the number line provides a clear visual in which all numbers less than -5 are shown in relation to this critical value.

To properly represent the inequality on the number line, you would start by marking the point corresponding to \( -5 \). Since \( -5 \) is not included in the solution, you would leave that point as an open circle. Then, you would shade or draw a line towards the left from \( -5 \) to show that all values lower than \( -5 \) satisfy the inequality. This graphically communicates that the solution includes an infinite set of numbers stretching to the left of \( -5 \) on the number line.