Solving Inequalities 🧮
What is an Inequality?
An inequality is like an equation, but instead of an equal sign (=), it uses symbols like > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). It shows that two values are not equal and helps us describe a range of possible answers. We use them in real life for things like speed limits ("drive less than 65 mph") or age restrictions ("you must be at least 13 to sign up").
How to Solve an Inequality
- Isolate the Variable: Use inverse operations (like addition/subtraction or multiplication/division) to get the variable by itself on one side of the inequality sign, just like you would with an equation.
- Special Rule: If you multiply or divide both sides by a negative number, you MUST flip the inequality sign. This is the most important rule to remember!
- Check Your Solution: Pick a number from your solution set and plug it back into the original inequality to see if it makes a true statement.
Worked Examples
Example 1: Solve x + 7 > 12
- Subtract 7 from both sides: x + 7 - 7 > 12 - 7
- Simplify: x > 5
✅ The solution is all numbers greater than 5.
Example 2: Solve -3y ≤ 15
- Divide both sides by -3 to isolate y: -3y / -3 ≤ 15 / -3
- FLIP THE SIGN! because we divided by a negative: y ≥ -5
✅ The solution is all numbers greater than or equal to -5.
Common Mistakes to Avoid ⚠️
- Forgetting to Flip the Sign: This is the #1 mistake! Always flip the inequality sign when you multiply or divide by a negative number.
- Confusing the Symbols: Remember, the "alligator mouth" > always opens towards the larger number. For "or equal to" (≥, ≤), the bar is underneath.
- Incorrect Graphing: Use an open circle for > or < and a closed circle for ≥ or ≤ when graphing the solution on a number line.
Tips & Tricks
- The Negative Number Flip: Think of it as the inequality getting "dizzy" and flipping over when it gets multiplied or divided by a negative.
- Check with a Number: After you solve, test your answer with a number that fits. For x > 5, try 6. Does 6 + 7 > 12? Yes! 13 > 12 is true.
- Equation First: If you're unsure, try solving it as an equation first (with an = sign), then put the inequality sign back in, remembering to flip it if you divided/multiplied by a negative.
How to Practice
To get better at solving inequalities:
- Start with simple one-step inequalities (like x - 3 < 10).
- Move on to two-step inequalities (like 2x + 5 ≥ 17).
- Always include problems that require you to flip the sign.
- Practice graphing your solutions on a number line.
- Create your own word problems. For example, "If I have $20 and toys cost $4 each, how many can I buy?" leads to the inequality 4t ≤ 20.