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Inequalities are mathematical expressions that show the relationship between two values where they are not equal. The symbols used in inequalities are: <, >, ≤, and ≥. They tell you if one value is less than, greater than, less than or equal to, or greater than or equal to another.

For example, in the inequality \(2x > -10\), it means that twice the value of \(x\) is greater than -10. When solving inequalities, similar to solving equations, you perform operations to isolate the variable. However, it's important to remember that multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign.

Knowing how to solve and graph inequalities will help you understand the range of possible solutions rather than a single value.

Interval notation is a shorthand used to write subsets of the real numbers. It is very handy for expressing solutions to inequalities. In interval notation:

• Parentheses \((a, b)\) indicate that the endpoints \(a\) and \(b\) are not included
• Brackets \([a, b]\) indicate that the endpoints \(a\) and \(b\) are included

For instance, the solution to \(x > -5\) is written in interval notation as \((-5, \, \infty)\). Here, the round parenthesis around -5 shows that -5 is not included in the solution, and \(\infty\) (infinity) is always used with a parenthesis since infinity is not a finite number that can be included.

Interval notation simplifies the way we represent and communicate our answers for ranges of values.

A number line is a visual representation of numbers in order. It helps to graphically portray the solution set of an inequality.

To graph the inequality \(x > -5\) on a number line, follow these steps:

• Draw a horizontal line and mark some reference points (numbers) on it.
• Locate -5 on the number line and draw an open circle around it. This indicates that -5 is not included in the solution (since the inequality is strict >, not ≥).
• Shade the region to the right of -5. This shows that all numbers greater than -5 are part of the solution.

By using a number line, you can easily see the range of values that satisfy the inequality. This visual aid is very useful, especially when dealing with more complex inequalities.