91Ó°ÊÓ

Inequalities are mathematical expressions involving the symbols >, <, ≥, and ≤, which compare the values of two expressions. In our example, the inequalities are:
- Inequality 1:
\(x + 1 > 3\)
- Inequality 2:
\(-4x + 1 > 5\)

Solving inequalities typically involves the same steps as solving equations, such as adding, subtracting, multiplying, or dividing both sides by a number. However, there's a crucial difference: when multiplying or dividing both sides by a negative number, the inequality sign must be reversed.
For example:
Dividing inequality 2 by -4 flips the > to a <, resulting in \(x < -1\).

It's also vital to interpret 'or' and 'and' conditions correctly:
- 'Or' means a value can satisfy either inequality.
- 'And' means values must satisfy both inequalities simultaneously.

Interval notation provides a way to describe sets of numbers and is particularly useful for expressing solutions to inequalities.

In our example:
The solution set \(x > 2\) is \( (2, \infty) \)
The solution set \(x < -1\) is \( (-\infty, -1) \)

The combined solution set using 'or' is \( (-\infty, -1) \cup (2, \infty) \).

Here’s a breakdown of how to read interval notation:
- Parentheses \( ( \) or \( ) \) indicate that the endpoint is not included (open).
- Brackets \( [ \) or \( ] \) indicate that the endpoint is included (closed).

Therefore, \( (2, \infty) \) means all values greater than 2, but not including 2, and extending indefinitely. Likewise, \( (-\infty, -1) \) means all values less than -1, but not including -1.

The union symbol \cup is used to combine different sets, indicating values that satisfy either condition.

Graphing inequalities on a number line visually represents the solution set, making it easier to understand which values satisfy the inequality.

In our example:

• Draw a number line.
• Plot the points of interest: 2 and -1.
• Use open circles at 2 and -1 since these values are not included in the solution set.
• Shade the region to the right of 2 without including 2 for \(x > 2\).
• Shade the region to the left of -1 without including -1 for \(x < -1\).

This graphical representation helps to see that the solution set extends to \( (2, \infty) \ and \ (-\infty, -1) \) without including the numbers 2 and -1 themselves.

Number lines are great tools to visualize ranges of values and show how they are combined in the case of compound inequalities.