Ever wonder why a simple “>” sign can split the whole number line into “yes” and “no” zones? That’s the magic of linear inequations.
💡 In Simple Words: A linear inequation is like a rule that tells you which numbers are bigger or smaller than a certain line. Solving it means finding all the numbers that follow the rule, and graphing shows those numbers on a number line.
What is a Linear Inequation?
A linear inequation (or inequality) looks just like a linear equation, but instead of an equal sign (=) it uses symbols like > (greater than),
For example, 3x – 5 > 7 means “all the values of x that make 3x – 5 bigger than 7.”
How to Solve a Linear Inequation – Step by Step
Solving is almost the same as solving an equation, except you have to watch out for one sneaky rule: if you multiply or divide by a negative number, you must flip the direction of the inequality sign.
Step 1 – Bring everything to one side
Move all terms so that the left side has the variable and the right side is a plain number.
Step 2 – Simplify
Combine like terms (numbers that have the same variable) just like you’d add up apples.
Step 3 – Isolate the variable
Divide or multiply until the variable stands alone. Remember the flip rule!
Step 4 – Write the answer in interval notation
Use parentheses ( ) when the endpoint is not included, and brackets [ ] when it is.
Graphing a Linear Inequation on a Number Line
Once you have the solution, you can draw it. Think of a number line as a road and the solution as the part of the road you’re allowed to drive on.
- Open circle – the endpoint is NOT allowed (for > or
- Closed circle – the endpoint IS allowed (for ≥ or ≤).
- Shade to the right for “greater than” and to the left for “less than”.
Worked Example 1
Solve and graph: 4x – 9 ≤ 7.
Step 1: Bring constants together – 4x ≤ 7 + 9 → 4x ≤ 16.
Step 2: Isolate x – divide by 4 (positive, so no flip) → x ≤ 4.
Solution in interval form: (‑∞, 4] .
Graph: draw a closed circle at 4 (because ≤ includes 4) and shade everything leftwards.
Worked Example 2
Solve and graph: -2x + 3 > 5.
Step 1: Move 3 to the right – -2x > 5 – 3 → -2x > 2.
Step 2: Divide by -2 (negative! flip sign) → x
Solution: (‑∞, -1) .
Graph: open circle at -1 (because > becomes
Quick Comparison – Solving vs. Graphing
| Action | Algebraic Step | Graphical Cue |
|---|---|---|
| Collect terms | Move everything to one side | Identify the “border” point |
| Isolate variable | Divide or multiply | Decide open or closed circle |
| Flip sign? | If you used a negative, reverse > to | Direction of shading changes |
| Write answer | Use interval notation | Shade appropriate side |
Common Mistakes to Avoid
- Forgetting to flip the inequality sign when dividing by a negative number.
- Mixing up open and closed circles on the graph.
- Leaving a stray “=” when the problem uses “>” or “
- Not checking the solution with a test value.
📝 Likely Exam Questions
- Solve 5x – 12 > 3x + 4 and express the answer on a number line.
Model answer: 5x – 12 > 3x + 4 → 2x > 16 → x > 8. Interval: (8, ∞). Open circle at 8, shade right. - Graph the solution of –3x + 2 ≤ 11.
Model answer: –3x ≤ 9 → x ≥ –3 (flip sign). Interval: [‑3, ∞). Closed circle at –3, shade right. - If the solution of 2x + 7
Model answer: 2(‑4) + 7 = –8 + 7 = –1, which is indeed less than 1. Hence the solution is correct. - Write the solution of 7 – x ≥ 2 in interval notation and draw the graph.
Model answer: –x ≥ –5 → x ≤ 5 (flip sign). Interval: (‑∞, 5]. Closed circle at 5, shade left. - Explain why the inequality sign flips when you divide by a negative number.
Model answer: Multiplying both sides of an inequality by a negative reverses the order of the numbers, just like turning a “greater than” sign upside‑down makes it “less than”.