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In mathematics, understanding inequality notation is crucial when solving problems that deal with relationships between quantities. It is the math equivalent of saying that two things are not equal in size, value, or quantity, and it is represented by symbols like '<', '>', '≤', and '≥'. Specifically, '<' means 'less than', '>' means 'greater than', '≤' means 'less than or equal to', and '≥' means 'greater than or equal to'.

When you see an inequality such as \(x < -3\), it indicates that the variable \(x\) can take on any value lower than -3 but not -3 itself. This is distinct from \(x ≤ -3\), which would include -3 as a possible value for \(x\). Grasping these symbols is fundamental for correctly interpreting and solving inequalities in algebra.

Graphing on a number line is a visual way to represent the solution set of an inequality and allows you to see all of the possible values the variable can be. To graph an inequality like \(x < -3\), you would start by drawing a horizontal line which represents the number line. You then locate the point that corresponds to the inequality's boundary, in this case, -3.

The next step is crucial: since \(x\) is less than -3 and not equal to -3, you would place a hollow circle, also known as an open circle, at -3. If \(x\) were less than or equal to -3 (\(x ≤ -3\)), you would use a filled-in circle to include -3 as part of the solution. Lastly, draw an arrow from the circle to the left for '\(<\)' (to indicate all numbers less than -3) or to the right for '\(>\)' (to indicate all numbers greater than -3).

Algebraic manipulation involves rearranging and simplifying algebraic expressions to solve for unknown variables. In the context of solving inequalities, the same principles that apply to equations hold true, with the addition of special rules related to the direction of the inequality sign. The process typically involves isolating the variable of interest on one side of the inequality.

For instance, with the inequality \(7x < -21\), you'd divide both sides by 7 to isolate \(x\). It's essential to remember that dividing or multiplying both sides of an inequality by a negative number reverses the inequality's direction. This rule does not apply in our example as we are dividing by a positive number. After dividing, simplification provides \(x < -3\), revealing the solution to the inequality. This step is necessary to not only find the solution but also to prepare it for graphing on a number line.