Linear Inequalities
Find the boundary, preserve the direction, and interpret which side of the boundary fits the condition.
Linear Inequalities
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An inequality describes a region of possible values—not just one answer
Equations usually identify exact values. Inequalities describe values below, above, or between boundaries. Solve them like equations, with one crucial rule: multiplying or dividing by a negative reverses the inequality sign.
≥ includes the boundary.
≤ includes the boundary.
> or < excludes the boundary.
Reverse the sign.
Boundary first
Solve the related equality to locate where the solution region begins or ends.
Context last
If the variable counts people, tickets, or boxes, choose a whole number that satisfies the inequality.
Solve, flip if negative, then test the direction
Example: Solve − 3x + 4 ≤ 16.
Why did ≤ become ≥ ? Dividing by − 3 reverses the order. Check x = 0: the original statement becomes 4 ≤ 16, which is true, so the solution must include 0. That confirms x ≥ −4.
Why it works
Testing one easy value catches the most common mistake: forgetting to reverse the inequality after dividing by a negative.
Five forms you should recognize
A van holds at most 900 kg. The driver is 70 kg and each box is 46 kg.
70 + 46b ≤ 900“At most” means the combined weight may equal 900 but cannot exceed it. Solving gives b ≤ 18.04…, so the greatest possible whole number is 18 boxes.
Subtracting does not reverse the sign. Dividing by − 3 does.
A closed point at − 2 means − 2 is included. Shading to the right means values greater than − 2.
x ≥ −2“More than − 1 but no greater than 5” contains two conditions that must both hold:
−1 < x ≤ 5The left endpoint is excluded; the right endpoint is included.
Tickets cost ₱85 and the budget is at most ₱600.
85t ≤ 600 → t ≤ 7.05…Because tickets are whole, 7 is possible but 8 is not. The greatest number is 7 tickets.
Check before you commit
- Forgetting to reverse the sign after dividing by a negative
- Reversing the sign after ordinary addition or subtraction
- Confusing “at least” with “at most”
- Using an open point for ≤ or ≥
- Rounding a whole − number maximum upward
- Treating “and” like “or” in compound inequalities
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Linear Inequalities FAQ
Why does the sign reverse with a negative?
Multiplying by − 1 reflects numbers across zero, reversing their order. For example, 2<5 but −2>−5.
How do I know whether to round up or down?
Return to the context. A minimum number of whole groups usually rounds up; a maximum number allowed usually rounds down.
Can an inequality have no solution?
Yes. Conditions such as x > 4 and x ≤ 2 have no overlap.
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