91Ó°ÊÓ

Compound inequalities are a way of expressing two inequalities that are often related to each other. Instead of solving them one by one, we take a look at them as a package. In our exercise, we have the compound inequality:
1. \(1 < 4 + 2x\)2. \(4 + 2x \leq 7\)
The goal of compound inequalities is to find a range of values for the variable, which in this case is \(x\), that satisfy all parts of the inequality simultaneously.

• Step 2 helps us solve the first part \(1 < 4 + 2x\) by simplifying it to \(-\frac{3}{2} < x\).
• Step 3 helps with the second part \(4 + 2x \leq 7\) simplifying it to \(x \leq \frac{3}{2}\).

We combine these solutions into one compound inequality: \(-\frac{3}{2} < x \leq \frac{3}{2}\). This tells us where \(x\) can safely exist between these two numbers, adhering to all parts of the inequality. It’s like setting boundaries for \(x\) on a number line!

Once we have a compound inequality, we can express its solutions in a compact form known as interval notation. This notation provides a clear and concise way of showing which numbers fit our solution.
To understand this, let's look again at the solution we've found: \(-\frac{3}{2} < x \leq \frac{3}{2}\).

• The round bracket \((-\frac{3}{2},\) signifies that \(-\frac{3}{2}\) is not included in the solution.
• The square bracket \(\frac{3}{2}]\), indicates that \(\frac{3}{2}\) is included.

Therefore, the interval \((-\frac{3}{2}, \frac{3}{2}]\) effectively summarizes all possible values of \(x\) that satisfy our compound inequality. It's like a shorthand for saying: "All numbers more than \(-\frac{3}{2}\) and up to and including \(\frac{3}{2}\) are in the solution set."

Graphing inequalities on a number line is a visual way to understand the solution set. It helps in clearly identifying which numbers are part of the solution and which are not.
To graph \((-\frac{3}{2}, \frac{3}{2}]\), follow these simple steps:

• Draw a number line.
• At \(-\frac{3}{2}\), place an open circle. This indicates \(-\frac{3}{2}\) isn't part of the solution.
• At \(\frac{3}{2}\), place a closed circle showing \(\frac{3}{2}\) is included in the solution set.
• Shade the area between the two points. This shaded region represents all the values that satisfy the inequality.

By graphing, you create a handy visual that makes it easy for anyone to quickly see the range of possible solutions. It's like a map that tells you where the values of \(x\) fit according to the compound inequality.