91Ó°ÊÓ

Inequalities are mathematical expressions that show the relative size of values. Unlike equations, which show equality, inequalities express relationships where one value is greater or less than another. To notate inequalities, we use special symbols:

• \(< >\) - Greater than
• \(< <\) - Less than
• \( \geq \) - Greater than or equal to
• \( \leq \) - Less than or equal to

When writing inequalities, it's crucial to understand whether the boundaries are inclusive or exclusive. If they are inclusive (i.e., the value is included in the range), we use the \( \leq \) or \( \geq \) symbols. If they are exclusive, we use \(<\) or \(>\) symbols. For example, in the inequality \(10 \leq y < 20\), 10 is included in the range as indicated by the \(\leq\) symbol, while 20 is not included, as shown by the \(<\) symbol.

Solving inequalities is similar to solving equations, with a few extra rules. The goal is to find the range of values for the variable that makes the inequality true. Here's the process broken down:

• Isolate the variable: Use algebraic operations to get the variable on one side of the inequality.
• Perform operations carefully: You can add, subtract, multiply, or divide both sides of an inequality by the same positive number without changing the inequality's direction. However, if you multiply or divide by a negative number, you must flip the inequality sign.
• Combine inequalities carefully: If you have multiple inequalities, consider them together to determine the overlapping range that satisfies all conditions.

For example, to solve \(2x + 3 < 15\), you would first subtract 3 from both sides to get \(2x < 12\). Then, divide by 2, yielding \(x < 6\). So, any value less than 6 satisfies the inequality.

Compound inequalities involve more than one inequality statement connected by the words 'and' or 'or'. These are often used to describe a range of values.

'And' Compound Inequalities: This means both conditions should be true simultaneously. For example, \( 1 < x \leq 4 \) means that \( x \) must be greater than 1 and less than or equal to 4. To solve, find the values that satisfy both conditions.

'Or' Compound Inequalities: This means either condition can be true. For instance, \( x < -2 \) or \( x > 3 \) indicates that \( x \) can be any value less than -2 or any value greater than 3. To solve, find the values that satisfy either condition.
An example from our exercise is the compound inequality \( 10 \leq y < 20 \). This particular inequality tells us that \( y \) must be at least 10 (inclusive) and less than 20 (exclusive). When solving such inequalities, it's critical to consider both parts to determine where they overlap.