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Solution set notation is a way of expressing the solutions to an inequality. It clearly shows which values satisfy the inequality. For the inequality \( x - 2 \geq -7 \), after solving for \( x \), we find \( x \geq -5 \). In solution set notation, this is expressed as:\[ \{ x \mid x \geq -5 \} \]This reads as "the set of all \( x \) such that \( x \) is greater than or equal to \(-5\)."
The vertical bar \( \mid \) means "such that," indicating the condition \( x \geq -5 \) must be met for values in the set.
Not only does this provide a clear mathematical expression, but it also aligns with set theory, showing the relationship between elements (in this case, numbers) and their membership in a set that meets a specific condition. This notation is widely used in mathematics for its clarity.

Graphing inequalities is a visual representation of the range of values that satisfy an inequality. To graph \( x \geq -5 \), we use a number line as a tool.First, identify the boundary of the inequality. Here, \( -5 \) is the boundary point for \( x \geq -5 \). Since \(-5\) is included in the solution (as indicated by \( \geq \)), we represent it with a solid dot on the number line.
Then, shade the portion of the number line that includes all values greater than \(-5\). This involves shading to the right of the point \(-5\).
The graph visually communicates the solution by illustrating both the starting point (\(-5\)) and the direction (right for "greater than or equal to"). Visualizing inequalities in this manner helps grasp the continuity and range of solutions.

The number line is a crucial aid in visualizing numerical relationships and inequalities. When working with inequalities like \( x \geq -5 \), the number line helps by offering a straightforward way to plot these solutions.When representing \( x \geq -5 \), you start by marking \(-5\) with a solid dot, indicating that this value is part of the solution.
The number line then extends to the right, illustrating all numbers greater than \(-5\) that satisfy the inequality.

• A solid dot indicates that the boundary point is included in the solution set (due to \( \geq \) or \( \leq \)).
• A hollow dot, conversely, would indicate exclusion of the boundary point (for \( > \) or \( < \)).

Using a number line simplifies understanding of which numbers fit an inequality, making such mathematical concepts accessible and easier to interpret.