Linear Inequality Solutions – A First Look
Imagine you're standing on a number line. You know exactly where the number 5 is. But what if I asked you to stand on "all numbers greater than 5"? You can't stand on all of them at once — they stretch infinitely to the right. That's the core idea of an inequality: instead of one exact point, you get a whole region of possible values.
A linear inequality is just like a linear equation (ax+b=0), but instead of an equals sign, you have one of these: <, >, ≤, or ≥. The solution is not a single number — it's an interval (or a union of intervals) on the number line.
From Equation to Inequality
Start with a simple equation:
Solve it: 2x=4⟹x=2. One point.
Now change it to an inequality:
Solve it the same way — but the meaning changes. Subtract 3: 2x>4. Divide by 2: x>2.
The solution is all numbers greater than 2. On a number line, you draw an open circle at 2 (because 2 itself is not included) and shade everything to the right.
If you multiply or divide both sides of an inequality by a negative number, the inequality sign reverses. For example: −x<5 becomes x>−5. This is the single most common mistake students make.
The Four Types of Solutions
| Inequality | Meaning | Number line representation |
|---|
| x>a | All numbers strictly greater than a | Open circle at a, shade right |
| x≥a | All numbers greater than or equal to a | Closed (filled) circle at a, shade right |
| x<a | All numbers strictly less than a | Open circle at a, shade left |
| x≤a | All numbers less than or equal to a | Closed circle at a, shade left |
The solution set is usually written in interval notation:
- x>2 → (2,∞)
- x≤−3 → (−∞,−3]
Parentheses ( or ) mean the endpoint is not included. Brackets [ or ] mean it is included.
Solving a Linear Inequality: Step by Step
Solve 3x−5≤7x+3.
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Bring variable terms to one side:
3x−5−7x≤3
−4x−5≤3
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Isolate the variable term:
−4x≤8
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Divide by the coefficient (here it's −4, so reverse the sign):
x≥−2
The solution is x≥−2, or in interval notation: [−2,∞).
Always check your answer by testing a number from the solution set. For x≥−2, test x=0: 3(0)−5=−5 and 7(0)+3=3. Is −5≤3? Yes. Now test a number outside, say x=−3: 3(−3)−5=−14 and 7(−3)+3=−18. Is −14≤−18? No — so the inequality fails, confirming our solution is correct.
Why This Matters …