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Graphing inequalities is a way to visually represent the solutions of inequalities on a number line. When we solve an inequality like \( x > 14 \), we are determining all the possible values of \( x \) that satisfy this condition. This is where the concept of an open circle and direction arrow come into play.

• Open Circle: When an inequality is strict, meaning it does not include equality (just like in \( x > 14 \)), we use an open circle. This indicates that the number itself is not included in the solution. Essentially, the open circle represents that \( 14 \) is not part of the solution set.
• Arrow Direction: The direction of the arrow on the number line shows whether \( x \) includes values larger or smaller than the number. For \( x > 14 \), the arrow moves to the right, encompassing all numbers greater than 14.

Graphing inequalities is crucial because it offers a clear and intuitive visual that accompanies the algebraic solution, making it easier to understand and check the results.
A well-drawn graph helps identify the range of possible values, confirming the inequality solution.

Isolation of a variable is a critical step in solving inequalities, similar to handling equations. The ultimate goal is to have the variable alone on one side of the inequality sign to determine its solution set. Let's break down how this is achieved.

• Identify the Variable: First, pinpoint the variable you need to solve for—in our case, it's \( x \).
• Undo the Operation: Next, you need to "undo" any operations that are being carried out on the variable. For the inequality \( x - 4 > 10 \), subtracting 4 from \( x \) is the operation that has to be reversed, so we add 4 to both sides.
• Perform the Operation: Add 4 to both sides: \( x - 4 + 4 > 10 + 4 \). This simplifies to \( x > 14 \).

Isolation efficiently simplifies the inequality, preparing it for further analysis or representation.
It ensures that the inequality is in its simplest form, helping in accurately graphing and verifying the solution.

Using a number line to represent solutions of inequalities is both informative and simple. A number line provides a clear, visual way of understanding the solutions in a straightforward manner. Here's how you can depict the inequality \( x > 14 \) on a number line:1. **Draw the Line**: Sketch a horizontal line and label it correctly with numbers.2. **Marking the Point**: Locate and mark the specific number involved in the inequality—here, it's 14.3. **Open Circle**: Place an open circle on 14. This symbolizes that 14 is not included in the solution.4. **Arrow Direction**: Since \( x \) must be greater than 14, draw an arrow to the right of the open circle. This indicates all numbers exceeding 14 are part of the solution.This representation technique allows one to instantly see and understand the range of numbers that satisfy the inequality.
It eliminates confusion over which numbers fulfill the inequality, offering an elegant graphical solution.