91Ó°ÊÓ

Interval notation is a way to represent the set of solutions of an inequality. It uses brackets and parentheses to describe which numbers are included in the solution set.

If an interval includes a number, we use square brackets [ ] around that number. If it does not include it, we use parentheses ( ).

For example, if we have an inequality like \( x > -3 \), this means all numbers greater than -3 are included, but -3 itself is not. In interval notation, this is written as \((-3, \infty)\). The parenthesis around -3 means -3 is not included, and the infinity symbol (\( \infty \)) indicates that the interval extends indefinitely.

Similarly, for \( x < -6 \), it means all numbers less than -6 are included, but not -6 itself. In interval notation, it is written as \((-\infty, -6)\). When combining these two intervals with an 'or' condition, we have two separate intervals: \((-\infty, -6) \cup (-3, \infty)\).

The \(\cup\) symbol stands for 'union' and represents that the solution set includes numbers from both intervals.

When graphing inequalities on a number line, we show which numbers are solutions by shading parts of the line.

For the inequality \( x > -3 \):

• Draw an open circle at -3 to indicate that -3 is not part of the solution.
• Shade the part of the number line to the right of -3, as it represents all numbers greater than -3.

For the inequality \( x < -6 \):

• Draw an open circle at -6 to indicate that -6 is not part of the solution.
• Shade the part of the number line to the left of -6, as it represents all numbers less than -6.

Using different segments of the number line, we visually represent where the solutions for each part of the compound inequality lie.

Solving inequalities involves finding the set of values of the variable that make the inequality true. Here’s how:

Step 1: Solve each inequality separately.

• For \( -x < 3 \), multiply both sides by -1 (and remember to flip the inequality sign): \( x > -3 \).
• For \( x < -6 \), the inequality is already solved and simplified.

Step 2: Combine the solutions.
The given problem is a 'compound inequality' with an 'or' condition, which means that either inequality can be true for any value to be a solution:

• If \( x > -3 \), then the interval is \((-3, \infty)\).
• If \( x < -6 \), then the interval is \((-\infty, -6)\).

Thus, the combined solution in interval notation is \((-\infty, -6) \cup (-3, \infty)\).

Understanding these steps ensures you can solve compound inequalities and express their solutions accurately.