Problem 13 - Algebra A Combined Function

Question

Solve each inequality. Graph the solution set. Write each answer using solution set notation. $$ x-2 \geq-7 $$

Step-by-Step Solution

Answer

\( x \geq -5 \); Graph: Solid dot on -5, shade to the right.

Step 1: Isolate the variable

To isolate the variable \( x \), we need to undo the subtraction of 2. We do this by adding 2 to both sides of the inequality:\[ x - 2 + 2 \geq -7 + 2 \] This simplifies to:\[ x \geq -5 \]

Step 2: Graph the inequality on a number line

To graph \( x \geq -5 \) on a number line, draw a solid dot on \( -5 \) because \( -5 \) is included in the solution set (indicated by \( \geq \)). Then, shade the number line to the right of \( -5 \) to represent all numbers greater than or equal to \( -5 \).

Step 3: Write in Solution Set Notation

The solution set notation for the inequality \( x \geq -5 \) is written as:\[ \{ x \mid x \geq -5 \} \]This reads as "the set of all \( x \) such that \( x \) is greater than or equal to \( -5 \)."

Key Concepts

Solution Set Notation

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

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.

Number Line Representation

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.