91Ó°ÊÓ

Graphing inequalities involves illustrating the set of solutions to an inequality on a number line or coordinate plane. When graphing simple linear inequalities like in our exercise, the number line is often used.
To graph an inequality such as \( x \leq -16 \), start by identifying the critical point, which is the boundary of the inequality. In this example, the critical point is \(-16\). On a number line, \(-16\) is marked with a closed circle to indicate that this point is included in the solution set.
After marking the point, draw a line from the closed circle extending towards negative infinity, showing all numbers that satisfy \( x \leq -16 \).

• Closed Circle: Used when the inequality symbol is "\( \leq \)" or "\( \geq \)," indicating the number is part of the solution.
• Open Circle: Used for "\( < \)" or "\( > \)" where the number is not included in the set.

Solving inequalities is a process much like solving equations, but with special attention to the direction of the inequality sign. The main goal is to isolate the variable on one side of the inequality.
For instance, take the inequality \( 7 + x \leq -9 \). The first step is isolating \( x \), which involves subtracting \( 7 \) from both sides of the inequality: \( x \leq -9 - 7 \). This helps remove constants that are added or subtracted from \( x \).
Once the variable is isolated, simplify the right-hand side. Here, \( -9 - 7 \) simplifies to \( -16 \), leaving us with the inequality \( x \leq -16 \).
While solving, if you multiply or divide both sides by a negative number, remember to reverse the inequality sign.

• Isolate the Variable: Use inverse operations to get the variable alone.
• Direction of Inequality: Multiplying/dividing by a negative necessitates reversing the inequality sign.

A number line is a simple yet powerful tool to visually represent solutions to inequalities.
It helps identify which numbers are included in a solution set and highlights the direction of the inequality. In the exercise, \( x \leq -16 \) is represented on a number line.
Start by drawing a horizontal line, which represents all real numbers. Identify \(-16\) on the line and mark it based on the inequality type: a closed circle for \( \leq \) or \( \geq \), and an open circle for \( < \) or \( > \).
Once the point is marked at \(-16\), extend a line to the left, indicating that all numbers to the left of \(-16\) are included in the solution. This extension visually confirms the solution as all numbers \( \leq -16 \).

• Identify Points: Highlight the boundary points of the solution set.
• Direction of Inequality: The shading direction indicates the solution's span.