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A number line is a visual tool used to represent numbers and their relationships. For inequalities, the number line helps show the range of values that satisfy an inequality. To sketch a number line, you draw a horizontal line and mark numbers at equal intervals. You can include zero and positive and negative numbers. These numbers are reference points that help us easily identify where other numbers lie.
To represent integers or specific values, you can use labeling. A number line can also illustrate if a number is included in a range or not. For instance, if a certain number is a boundary but not part of the solution, we represent it with an open circle above the number. If the number is part of the solution, we use a closed circle. This visual representation makes it intuitively clear whether particular values are part of the solution set.

Solving inequalities is a crucial skill in mathematics, and it's quite similar to solving equations. The goal is to isolate the variable on one side. Let's take the inequality \(4x < -12\) as an example. Here’s how to solve it:

• Firstly, identify the operation being performed on the variable. In this case, \(x\) is being multiplied by 4.
• To isolate \(x\), perform the inverse operation. Divide both sides of the inequality by 4, resulting in \(x < -3\).

Once solved, the inequality shows that \(x\) can be any number less than -3. Solving inequalities involves similar steps to equations, but keep in mind the direction of the inequality sign, especially if multiplication or division by negative numbers are involved, as these reverse the inequality sign.

Inequality notation is a way to express the size relationship between two values or expressions. When we write \(x < -3\), it means that \(x\) is less than \(-3\).
It's important to understand the different symbols used in inequality notation:

• <: This symbol means "less than." For example, \(a < b\) reads as "a is less than b."
• \(\leq\): This symbol means "less than or equal to." It includes equality, such as \(a \leq b\) reads as "a is less than or equal to b."
• >: This symbol means "greater than." For example, \(a > b\) means "a is greater than b."
• \(\geq\): This symbol means "greater than or equal to." It also covers equality, such as \(a \geq b\) reads as "a is greater than or equal to b."

These symbols help quickly identify which values are included or excluded from the solution set. Always pay attention to the symbols as they directly influence how you represent the solution on a number line or in interval notation.