Problem 39 - College Algebra with Modeling and Visualization

Question

Solve the inequality graphically. Use set-builder notation. $$ x+2 \geq 2 x $$

Step-by-Step Solution

Answer

The solution in set-builder notation is \( \{ x \mid x \leq 2 \} \).

Step 1: Rearrange the Inequality

Start by rearranging the terms of the inequality by moving all terms involving the variable to one side. Subtract \(x\) from both sides: \(x + 2 - x \geq 2x - x\). We then simplify to get \(2 \geq x\).

Step 2: Identify Critical Point

The critical point of the inequality \(2 \geq x\) is \(x = 2\). This value will be important for plotting and testing regions on a number line.

Step 3: Plot the Critical Point on a Number Line

Draw a number line and mark the critical point \(x = 2\). Since the inequality includes "greater than or equal to", mark \(x = 2\) with a closed dot to indicate that \(x\) can equal 2.

Step 4: Determine the Solution Region

Since \(2 \geq x\) implies \(x\) can be any value less than or equal to 2, shade the region to the left of \(x = 2\) and include \(2\) itself on the number line.

Step 5: Write the Solution in Set-Builder Notation

Transform the shaded region into a set using set-builder notation, which describes the solution set. The solution is \( \{ x \mid x \leq 2 \} \), meaning all values of \(x\) that are less than or equal to 2.

Key Concepts

Understanding Set-Builder Notation

Set-builder notation is a helpful way to describe a set of numbers that fulfill a particular condition, especially in regard to inequalities. It provides a concise way to represent a solution set without having to list out all possible numbers.
For example, consider the inequality provided: \( x + 2 \geq 2x \). After rearranging and simplifying, we have \( 2 \geq x \).
In set-builder notation, this can be written as \( \{ x \mid x \leq 2 \} \). Let's break this down:

• \( x \) stands for the variable.
• The vertical bar \( | \) translates to "such that".
• \( x \leq 2 \) describes the condition that \( x \) must satisfy.

Essentially, the notation means "the set of all \( x \) such that \( x \) is less than or equal to 2." With practice, you'll find set-builder notation is an efficient tool in mathematical communication.

Visualizing Solutions with a Number Line Graph

A number line graph is a visual tool that makes understanding inequalities more intuitive. When solving \(2 \geq x\), you are asked to find all values of \( x \) that satisfy this condition and represent them on a number line.
First, draw a horizontal line to serve as your number line. Then, identify key points, or critical points, relevant to the inequality. For \( 2 \geq x \), we need to be clear about where \( x \) can be.

• Locate the number \( 2 \) on the line.
• Because \( x \) can be exactly \( 2 \), use a closed dot to mark this point.
• The inequality states \( x \) is less than or equal to \( 2 \), so shade the line to the left of \( 2 \).

This shaded region represents the solution, encapsulating all possible values of \( x \) that make the inequality true. The use of shading and open or closed dots helps clarify which values are included or excluded from the solution set.

Critical Point Identification and its Importance

When solving inequalities, identifying critical points is crucial. A critical point, like the \( x = 2 \) in this problem, represents a boundary between different regions of solutions.

• For the inequality \( 2 \geq x \), the critical point separates values where the inequality holds true from those where it does not.
• It's represented distinctly on a number line by a closed or open dot, indicating whether the point itself is included in the solution.

The critical point is determined by turning the inequality into an equation (i.e., \( 2 = x \)). This solution serves as the deciding factor for defining solution areas on the number line.
Understanding how to identify and represent critical points helps in accurately graphing inequalities and finding the correct solution regions. Without this step, it would be challenging to visually distinguish which solutions are valid.