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Inequality graphing is a visual way to represent the solutions to an inequality. It helps you see at a glance what values of the variable satisfy the inequality. For example, consider the inequality solved in our exercise:
When graphing the solution to an inequality, such as \(x \geq -2\), we start by drawing a number line. After marking the point \(-2\) on the number line, we determine whether to use a solid dot or an open circle. Since the inequality includes the equal sign (\(\geq\)), we use a solid dot to indicate that \(-2\) is part of the solution set.

Next, because the inequality is \(x \geq -2\), all numbers to the right of \(-2\) on the number line are also solutions. To represent this, we draw a line extending to the right from the solid dot, often extending to an arrow to show that the line continues indefinitely. This visual representation considerably aids in understanding the set of possible solutions and is a crucial technique to master.

The number line is a fundamental tool in mathematics for showing the order of numbers and representing intervals or sets of solutions. Representing inequalities on a number line allows students to understand the range of values that satisfy the inequality.

For the solution \(x \geq -2\), we place a solid circle on \(-2\) to show it is included in the range of numbers that make the original inequality true. A ray extending to the right, often with an arrow at the end, represents all the numbers greater than \(-2\). The direction of the arrow encapsulates infinity in the respective direction, symbolizing the continuation of all numbers beyond what we can concretely draw or represent.

Importance of Solid and Open Circles

A solid circle is used when a number is a part of the solution, while an open circle is used when a number is not part of the solution. This difference is crucial as it directly affects which numbers are considered valid solutions.

Algebraic manipulation is the process of rearranging and simplifying algebraic expressions and equations, which often involves a series of steps to solve for a variable. This process is critical in solving inequalities effectively. Let's explore the steps taken in the original exercise:

First, we simplified the inequality by subtracting \(3\) from both sides, leading to \(-7x \leq 14\). This is an example of applying the subtraction property of equality, which states that subtracting the same number from both sides of an inequality does not change its solutions.

As a next step, we divided both sides by \(-7\) to solve for \(x\). This action demonstrates an important rule in solving inequalities: when both sides of an inequality are multiplied or divided by a negative number, the direction of the inequality must be reversed. This reversal occurs because multiplying or dividing by a negative number changes the order of the numbers on the number line. Thus, after division, we get \(x \geq -2\), the final form of the solution we were seeking.