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Graphing inequalities helps to visually represent solutions where a variable satisfies a certain condition. In this case, we had the inequality \(y \geq -5\). Once solved, inequalities can be graphed easily on a number line, which is a simple, one-dimensional representation that captures the range of possible solutions. To graph \(y \geq -5\), begin by marking the point -5 on the number line. The sign "greater than or equal to" tells us that -5 is included in the solution set. We represent inclusion by drawing a solid circle over this point. This visual method confirms -5 is indeed a solution.Then, since the inequality specifies numbers greater than -5, draw a continuous line or arrow pointing to the right. This arrows away from -5 reflects all the numbers that satisfy the condition \(y \geq -5\). This graphical method provides clarity and makes verifying solutions easier.

Number line representation transforms abstract algebraic solutions into clearer visual diagrams. By plotting solutions to inequalities on a number line, we can see which numbers make the inequality true. When placing a solution point, it's crucial to determine whether to use an open or solid circle.

• A solid circle is used when the number is included in the solution (e.g., for \(\geq\) or \(\leq\) signs).
• An open circle indicates that the number itself is not part of the solution (used for \(>\) or \(<\) signs).

This visual distinction is important as it immediately shows whether specific endpoints are included or excluded.In our example with \(y \geq -5\), the solid circle at -5 ensures that -5 itself is a solution, while the arrow illustrates numbers extending infinitely in the positive direction are also valid solutions.

Algebraic manipulations allow solving inequalities by simplifying or rearranging the terms to isolate the variable. This technique shares similarities with solving equations, however, special care is needed since inequalities have distinct behaviors.To isolate the variable "y" in the original inequality \(-3 \leq y + 2\), we begin by eliminating constants affecting the variable. Subtraction or addition achieves this simplification. For instance, subtracting 2 from both sides simplifies to \(-3 - 2 \leq y + 2 - 2\), resulting in \(y \geq -5\).It’s important to remember that when multiplying or dividing both sides of an inequality by a negative number, the inequality sign flips. Without this step, the inequality may not correctly represent the solution set. Fortunately, in this example, that step was not necessary. Properly approaching algebraic manipulations ensures precise solution expressions and clarity when continuing to graph the result.