Why linear inequalities matter in everyday life
Ever wondered how a shop decides "buy 3 or fewer items for a discount"? That's a real‑world linear inequality at work.
In simple words, a linear inequality is like a balance scale that tells you which numbers make one side bigger, smaller, or equal to the other. Solving it means finding all those numbers, and graphing shows them on a line.
What is a linear inequality?
A linear inequality looks just like a linear equation (something like 2x+3=7) but instead of an equals sign (=) it uses < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). The word "linear" means the highest power of the variable (usually x) is 1.
Key terms explained
- Inequality sign: The symbol that tells you the relationship (, ≤, ≥).
- Solution set: All the numbers that satisfy the inequality.
- Number line: A straight line marked with numbers, used to picture the solution set.
Step‑by‑step method to solve a linear inequality
Think of solving an inequality as a short recipe. Follow each step, and you’ll never miss a sign change.
- Write the inequality in standard form (all terms on one side, variable on the left).
- Isolate the variable by adding or subtracting the same number on both sides.
- If you multiply or divide by a negative number, flip the inequality sign ( and vice‑versa).
- Express the answer as a range (for example, x > 4) or in interval notation.
Worked example 1: Solving
Solve 3x - 7 ≤ 2x + 5.
- Bring all x terms to one side: 3x - 2x ≤ 5 + 7 → x ≤ 12.
- No multiplication or division by a negative, so the sign stays.
- Solution set: all numbers less than or equal to 12.
Graphing a linear inequality on a number line
Take the result x ≤ 12. On a horizontal line, mark 12 with a solid dot (solid because ≤ includes 12). Shade everything to the left because the inequality wants numbers smaller than or equal to 12.
If the sign were < instead of ≤, you’d use an open circle to show that 12 itself isn’t allowed.
Quick reference table
| Step | What to do | Watch out for |
|---|---|---|
| 1 | Move terms so variable is alone on left | Keep the inequality sign the same |
| 2 | Add or subtract the same number both sides | Sign never changes here |
| 3 | Multiply or divide both sides | If the number is negative, flip the sign |
| 4 | Write solution set | Use ≤ or ≥ for inclusive, for exclusive |
| 5 | Draw on number line | Solid dot for ≤/≥, open circle for |
Common mistakes to avoid
- Forgetting to flip the sign when dividing by a negative number.
- Leaving the variable on the right side and then drawing the graph backwards.
- Using a solid dot for a strict inequality ().
📝 Likely Exam Questions
- Solve and graph: 5 - 2x > 9.
Answer: -2x > 4 → x . Open circle at -2, shade left. - Find the solution set of: 4x + 1 ≤ 3x - 6.
Answer: x ≤ -7. - Explain why the sign must be reversed when multiplying an inequality by -3.
Answer: Multiplying by a negative flips the order of numbers; to keep the statement true, the inequality sign reverses. - Graph the inequality: x > 0.5 on a number line.
Answer: Open circle at 0.5, shade right. - Combine two inequalities: -2 ≤ 3x - 4 and give the final range for x.
Answer: Add 4: 2 ≤ 3x ; divide by 3: 2/3 ≤ x .