91Ó°ÊÓ

A compound inequality is like having two mail packages instead of one. It contains two separate inequalities joined by either "and" or "or." In our exercise, the compound inequality was \( -4 < 3x + 2 < 5 \). This is an "and" situation, meaning we are looking for values of \( x \) that fit both restrictions simultaneously.

Start by breaking it down. Imagine two statements: \(-4 < 3x + 2\) and \(3x + 2 < 5\). You solve each part on its own to uncover the possible values for \( x \). When both parts are resolved, as in this example, \(-2 < x\) and \(x < 1\), you have found a range of numbers that satisfy the entire statement.

This type of problem is great for practicing problem-solving skills because it requires finding common ground in two different scenarios.

Interval notation is like using a shortcut to show a list of all the possible solutions for \( x \). It's a streamlined way to express a range of numbers, especially useful when describing solutions of inequalities.

In our example, the final result of the compound inequality is the range \(-2 < x < 1\). In interval notation, this is written as \((-2, 1)\).

Some key things to remember:

• A parenthesis \((\) means the endpoint is not included, like an open circle in graphing.
• A square bracket \([\) means the endpoint is included, marked by a closed circle.

You will use parentheses here because the numbers -2 and 1 are not part of the solution themselves.

Interval notation helps when summarizing solutions so they are easy to read and understand.

Graphing inequalities is like turning a math problem into a picture. It helps visualize the range of solutions on a number line. In the exercise, after solving \(-2 < x < 1\), you draw this on a number line to represent which numbers work.

For our example:

• Place open circles at -2 and 1, indicating these numbers are not part of the solution but are limits.
• Shade the segment between the circles because all numbers in this space satisfy the inequality.

Remember, an open circle means "not included," and a closed circle would mean "included."

Graphing inequalities helps convert algebraic solutions into a visual format, making it easier to verify the correctness of solved inequalities.