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Graphing inequalities on a number line is a visual way to showcase the solution set of an inequality. A number line is simply a straight, horizontal line where numbers are marked at even intervals. Think of it like a ruler, but instead of measuring in inches or centimeters, it shows values of numbers.

For the inequality \(x < -2\), you will begin by preparing your number line. First, draw a horizontal line and decide on a scale, usually using whole numbers. Mark the key number from the inequality, in this case, -2. This point becomes your reference.

Since the inequality is asking for values of \(x\) that are less than -2, after marking -2, you'll draw an arrow that extends to the left from this point. The arrow indicates all the numbers less than -2. Alongside the arrow, draw a small circle at -2 that remains open, illustrating that -2 itself is not part of the solution. Once drawn, this number line clearly shows the set of numbers solving the inequality.

Solving an inequality is similar to solving an equation, but with a twist. Instead of finding the exact value of \(x\), in inequalities, you find a range of potential values.

To solve the inequality \(x < -2\), interpret it as asking: 'What numbers are smaller than -2?' The set of numbers meeting this criterion can be infinite and include any number to the left of -2 on a number line, such as -3, -4, -10, or even -2.5 or -100.

The solution to this inequality also needs to be expressed on a number line. The key here is understanding whether the boundary number, -2 in this example, should be included in the solution set or not. For \(x < -2\), we only want numbers smaller than -2, so -2 is excluded, signified by an open circle on the number line representation.

The 'less than' symbol, denoted as \(<\), is quite intuitive once you know what to look for. It is a key player in expressing inequalities. The symbol itself resembles an arrow pointing to the left, which can help you remember that it means looking for numbers that are smaller than the number on the other side of it.

For instance, in the inequality \(x < -2\), the \(<\) symbol tells us that the solution includes all numbers smaller than -2.
This also means -2 is not a part of the solutions, which is why we draw an open circle on the number line over -2. This open circle is a useful visual cue, signaling that we are not "counting" the number itself, just those to its left.

In summary, the less than symbol is vital in defining the solution set for inequalities, helping us visualize which numbers are included and marking those that are excluded.