An inequality describes a range of possible values rather than one required value. Solving it means preserving the order relationship while isolating the variable, then representing every valid value accurately.
Symbols and verbal language
| Relationship | Common language | Number-line endpoint |
|---|---|---|
| \(x<a\) | less than; below; fewer than | open at \(a\) |
| \(x>a\) | greater than; above; more than | open at \(a\) |
| \(x\le a\) | at most; no greater than; does not exceed; maximum \(a\) | closed at \(a\) |
| \(x\ge a\) | at least; no less than; minimum \(a\) | closed at \(a\) |
Open and closed endpoints
An open circle excludes its endpoint and matches \(<\) or \(>\). A closed circle includes its endpoint and matches \(\le\) or \(\ge\). The highlighted ray points toward all remaining solutions.
Four basic inequality graphs
Compare strict and inclusive endpoints and both ray directions.
Properties that preserve or reverse order
| Operation applied to both sides | Effect on inequality direction |
|---|---|
| Add or subtract any real number | direction stays the same |
| Multiply or divide by a positive number | direction stays the same |
| Multiply or divide by a negative number | direction reverses |
| Transitive reasoning: \(a<b\) and \(b<c\) | conclude \(a<c\) |
A negative coefficient
Solve \(-3(2x-5)>21\).
- Distribute
\(-6x+15>21\).
- Subtract
\(-6x>6\). Addition or subtraction does not reverse the sign.
- Divide by negative six
\(x<-1\). Reverse \(>\) to \(<\).
- Check
At \(x=-2\), the original left side is \(27\), which is greater than \(21\).
Multi-step and variable-on-both-sides cases
Reliable solving workflow
- Simplify
Distribute and combine like terms on each side.
- Collect the variable
Move variable terms to one side using addition or subtraction.
- Collect constants
Move constant terms to the other side.
- Isolate
Divide by the coefficient; reverse only if that divisor is negative.
- Interpret
Graph or apply the resulting range to the context.
Variables on both sides
Solve \(5-2x\le3x+20\).
- Subtract three x
\(5-5x\le20\).
- Subtract five
\(-5x\le15\).
- Divide by negative five
\(x\ge-3\).
Fractional coefficient
Solve \(\frac{3}{4}x-2>7\).
- Add two
\(\frac{3}{4}x>9\).
- Multiply by four thirds
\(x>12\). The multiplier is positive, so the sign stays.
Context, possible values, and extrema
Context may restrict values to integers or nonnegative quantities. If a whole-number count satisfies \(n<8\), its greatest possible value is \(7\). Without an integer restriction, there is no greatest real number less than \(8\).
A contextual maximum
A container holds at most \(54\) kilograms. It already holds \(18\) kilograms, and each package weighs \(4.5\) kilograms. Find the greatest whole number of packages.
- Model
\(18+4.5p\le54\).
- Solve
\(4.5p\le36\), so \(p\le8\).
- Apply the domain
Package count is a whole number, so the greatest possible count is \(8\).
Check your understanding
Solve \(-4x+7<19\).
- \(x<-3\)
- \(x>-3\)
- \(x<3\)
- \(x>3\)
Show answer and explanation
Answer: \(x>-3\)
Subtract \(7\) to get \(-4x<12\), then divide by \(-4\) and reverse the inequality.
What to remember
- Open endpoints match strict inequalities; closed endpoints include equality.
- Adding or subtracting never forces a reversal.
- Multiplying or dividing by a negative reverses the order relationship.
- Apply integer, whole-number, or nonnegative restrictions only when the context states them.
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.