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Understanding how to visually represent solutions is a vital part of solving inequalities. This process, known as inequality graphing, allows students to easily visualize the set of all possible solutions that satisfy the inequality.

When graphing the solution to an inequality, such as b > -1.5, a number line is used as the basis for the graph. You start by marking the critical value, in this case, -1.5, on the number line. Since -1.5 is not included in the solutions (indicated by '>' rather than '≥'), we represent it with an open circle. Then, because the inequality is true for all numbers greater than -1.5, an arrow is drawn from the open circle to the right, extending to infinity.

For inequalities, remember that an open circle is used when the variable is greater than (b > -1.5) or less than a number (b < -1.5), whereas a closed circle is used for 'greater than or equal to' (≥) or 'less than or equal to' (≤), signifying that the number in question is included in the solution set.

One key step in solving inequalities, and equations in general, is isolating the variable. This is the process of arranging the inequality so that the variable you are solving for (in this case, b) is on one side of the inequality sign, and all the other terms are on the opposite side. By doing this, the inequality simplifies down to a basic form that makes it easier to visualize and solve.

In the given exercise, isolating the variable was done in two steps. First, by subtracting 7.5 from both sides to remove the constant term from the side with the variable, and second, by dividing by -3. Remember, when dividing across an inequality by a negative number, it’s important to reverse the direction of the inequality sign to maintain the proper relationship. Isolating variables is a foundation of algebra and ensures that students can move towards the solution effectively.

The inequality solutions represent all values that satisfy the inequality after we solve it. It's critical to understand that unlike equations, inequalities often have a range of possible answers.

When solving the inequality -3b < 4.5, we divided each side by -3, paying attention to switch the inequality sign, resulting in b > -1.5. The solution to this inequality includes every number that is greater than -1.5, which is an infinite set. Demonstrating this visually on a number line helps to grasp, not just the concept of infinity, but also the practical application of inequalities in real-life situations where we deal with ranges or intervals rather than exact numbers. Keep in mind that verifying your solution is also vital. You can do this by plugging in values from your solution set back into the original inequality to check if they satisfy the condition set by the inequality.