Ever wondered how a simple “greater than” sign can decide where a basketball lands on the court?
In simple words, a linear inequation is like a rule that tells you which numbers are allowed and which are not, just like a “no‑entry” sign on a road.
What is a Linear Inequation?
A linear inequation (or inequality) is an expression that uses <, >, ≤ or ≥ instead of an equal sign. It compares two algebraic expressions and says one is bigger, smaller, or equal in size.
Key terms you’ll meet
- Inequality symbol: The sign <, >, ≤, or ≥ that shows the relationship.
- Solution set: All the numbers that make the inequality true.
- Number line: A straight line that shows all real numbers; we shade the part that satisfies the inequality.
Steps to Solve a Linear Inequation
Think of solving an inequation as a recipe. Follow each step, and you’ll end up with the correct “allowed” numbers.
Worked Example 1
Solve 3x - 5 < 7.
- Add 5 to both sides: 3x < 12.
- Divide by 3 (a positive number, so the sign stays the same): x < 4.
Solution set: all numbers smaller than 4. On a number line we draw an open circle at 4 and shade everything to the left.
Worked Example 2 (negative coefficient)
Solve -2x + 3 ≥ 9.
- Subtract 3 from both sides: -2x ≥ 6.
- Divide by -2. Because we divide by a negative, the inequality sign flips: x ≤ -3.
Solution set: all numbers less than or equal to -3. Here we use a closed circle at -3 and shade leftwards.
Graphing the Solution on a Number Line
After you have the final inequality (like x < 4), draw a horizontal line, mark the critical number (4), decide if the circle is open (strict < or >) or closed (≤ or ≥), and shade the appropriate side.
- Open circle = the endpoint is NOT included.
- Closed circle = the endpoint IS included.
- Shade left for “<” or “≤”, shade right for “>” or “≥”.
Quick Comparison: Solving vs. Graphing
| Aspect | Solving (Algebraic) | Graphing (Visual) |
|---|---|---|
| Goal | Find a formula like x < 4 | Show the region on a line |
| Key step | Isolate x, watch sign flips | Choose open/closed circle |
| Exam check | Write the correct inequality | Draw correct shading |
Common Mistakes to Avoid
- Forgetting to flip the inequality sign when dividing or multiplying by a negative number.
- Using a closed circle when the sign is strict (< or >).
- Leaving the variable on both sides after you’ve already isolated it.
- Mixing up which side to shade – remember “<” means left, “>” means right.
📝 Likely Exam Questions
- Solve and graph 5x - 2 ≤ 13.
Answer: x ≤ 3. Shade left of 3, closed circle at 3. - Find the solution set of -4x + 7 > 15.
Answer: -4x > 8 → x - Explain why the inequality sign reverses when you divide by a negative number.
Answer: Multiplying or dividing by a negative flips the order of numbers; e.g., -3 < -1, but dividing both by -1 gives 3 > 1. - Graph the solution of 2x + 5 ≥ 1 and write it in interval notation.
Answer: 2x ≥ -4 → x ≥ -2. Closed circle at -2, shade right. Interval: [-2, ∞). - Given the inequation 3 - x < 0, state the solution set and draw the number line.
Answer: -x < -3 → x > 3. Open circle at 3, shade right. Interval: (3, ∞).