91Ó°ÊÓ

Understanding number line representation is crucial when working with inequalities. It visually shows the range of possible solutions.
To sketch the solution set for the inequality \(-8 < x < -1\), start by drawing a horizontal line.
Mark the numbers -8 and -1 on this line.
Since the inequality symbols are '<' (less than) and not '≤' (less than or equal to), these endpoints are not included in the solution set.
Use open circles at -8 and -1 to indicate this.
Then, shade the region between these two points to show all the values that x can take (every number greater than -8 and less than -1).
This provides a clear, visual method to showcase the solution set of the inequality.

Compound inequalities involve two separate inequalities combined into a single statement, usually connected by 'and' or 'or'.
In our exercise, we have the compound inequality \(-3 < x + 5 < 4\), which implies two conditions:

• \(-3 < x + 5\)
• \(x + 5 < 4\)

To solve this, we need to make sure x satisfies both conditions simultaneously.
First, subtract 5 from all three parts to isolate x in the middle. This gives us \(-8 < x < -1\).
This tells us that x must be greater than -8 and less than -1 at the same time.
Compound inequalities like this one are fairly common, so mastering them will help you solve many different types of mathematical problems in the future.

Algebraic manipulation involves performing operations to simplify or solve equations and inequalities.
In our example, the inequality \(-3 < x + 5 < 4\) required us to isolate x.
Start by subtracting 5 from every part of the inequality:

• \(-3 - 5 < x + 5 - 5 < 4 - 5\)
• which simplifies to \(-8 < x < -1\)

This step-by-step manipulation allowed us to find the range of values for x.
Remember, whatever operation you do to one part of the inequality, must be done to all parts.
This preserves the equality or inequality relationship.
Simple operations like addition, subtraction, multiplication, and division (except by zero) are frequently used.
Mastering these manipulations will make it easier to solve more complex inequalities and equations.