Inequalities

While equations show balance, inequalities show comparison. Instead of saying two things are equal, inequalities tell us that one side is larger, smaller, or within certain limits. They use symbols like <, >, ≤, and ≥ to describe ranges of possible values. That means there isn’t always one single answer – there’s a whole set of them.

For example, 3x + 2 > 8 doesn’t pinpoint x; it describes a condition. Solving it step by step gives x > 2, meaning any number greater than 2 works. That’s a huge idea – algebra isn’t just about finding one result, but about describing possibilities. When you represent these on a number line, the open or closed circle shows whether the boundary is included or not, creating a visual of the solution’s range.

One key rule separates inequalities from regular equations: when you multiply or divide by a negative number, the inequality sign flips. For instance, if −2x < 8, dividing both sides by −2 gives x > −4. This rule maintains truth and consistency, just like balancing equations does.

In the real world, inequalities describe limits and conditions: speed restrictions, temperature ranges, safety margins, and budgets. They’re everywhere we compare quantities and say, “this must be less than that.” Learning to read and manipulate them turns abstract symbols into useful tools for decision-making and reasoning.

Hints and Tips:

• Treat inequalities like equations until multiplying or dividing by negatives – then flip the sign.
• Represent solutions clearly on a number line.
• Test values within and outside your range to confirm your result.
• Remember: inequalities describe many possible answers, not just one.