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Compound inequalities involve two or more inequalities that are combined into one statement. They often use 'and' or 'or' to link the inequalities.
In the example provided, the compound inequality is given as: \(6 \geq x \geq -4\). This chain of inequalities means the following:
- The value of \( x \) must be less than or equal to 6.
- The value of \( x \) must also be greater than or equal to -4.
Such inequalities are expressed compactly and capture a range of values that \( x \) can take.
When dealing with compound inequalities, it's crucial to remember:

• Identify both parts of the inequality.
• Solve each part separately if needed.
• Combine the solutions to ensure \( x \) satisfies the complete statement.

Interval notation is a way of writing the set of all solutions to an inequality. It uses brackets to denote the range and boundaries of the solution set.
Here are the key symbols used in interval notation:
- \([a, b]\) means \(a\) and \(b\) are included in the interval.
- \((a, b)\) means \(a\) and \(b\) are not included in the interval.
- \((a, b]\) means \(a\) is not included, but \(b\) is.
- \([a, b)\) means \(a\) is included, but \(b\) is not.
In the provided exercise, the compound inequality \(6 \geq x \geq -4\) translates to the interval notation \([-4, 6]\) because both surface values \(-4\) and \(6\) are included.
Always pay attention to whether brackets are open or closed to correctly indicate boundaries. Closed brackets [ ] mean inclusive, while open brackets ( ) mean exclusive.

A number line is a visual representation of numbers on a straight line. It helps in understanding and solving inequalities by showing the position of numbers.
Plotting an inequality on a number line involves several steps:
- Identify key points like boundaries or endpoints.
- Use solid or hollow circles to denote inclusive or exclusive boundaries, respectively.
- Shade the region that represents the solution set.
In the example, for the inequality \(6 \geq x \geq -4\) illustrated:

• Mark -4 and 6 on the number line with solid circles to denote they are included in the solution set.
• Shade the line between -4 and 6 to show all numbers in this range are part of the solution.

Using these visual aids makes it easier to comprehend which values satisfy the inequality.