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Isolating the variable is a fundamental technique in solving inequalities, just as it is with equations. The goal is to get the variable on one side of the inequality symbol to clearly see its possible values. For the given inequality, the process starts by removing or 'isolating' additional terms that are on the same side as the variable. In our example, we start with 5 + r ≥ -5. To isolate r, we need to eliminate the 5 on the left by subtracting 5 from both sides of the inequality. This yields the simpler inequality r ≥ -10. Through this process, not only have we isolated the variable, but we have also prepared it to be represented on a number line, making the solution set visible and understandable.

Algebraic operations include the fundamental processes of addition, subtraction, multiplication, and division which we can apply to both sides of an inequality without changing its direction. In the context of inequalities, it's crucial to remember that multiplying or dividing by a negative number will reverse the inequality sign. In the example 5 + r ≥ -5, we perform a subtraction operation, taking away 5 from both sides. This doesn't involve reversing the inequality because subtraction doesn’t have the same effect as multiplying or dividing by a negative number. By understanding and correctly applying algebraic operations, we can manipulate inequalities to reveal the solutions clearly.

Graphing an inequality on a number line helps to visually represent the range of possible solutions. After solving the algebraic inequality, such as r ≥ -10, the next step is to draw a number line. On this line, we mark the value that the variable is greater than or equal to, in this case, -10. As r includes -10, we use a closed circle or dot to indicate that -10 is part of the solution. The circle is then extended with a line or arrow to the right to represent all the numbers greater than -10 which are also included in the solution. This visual representation is intuitive and allows for quick comprehension of the range of valid values for the variable.

Representing inequalities on a number line provides immediate visual insight into the solution set. In the previously solved inequality r ≥ -10, the closed circle placed at -10 signifies that this number is included in the solution set. The line extending to the right signals that every number greater than -10 is also a solution. The number line method is a powerful tool to depict the continuous nature of inequalities – unlike discrete possibilities in equations – highlighting the infinitely many values that satisfy the inequality.