The Central Idea
An equation asks when two expressions are equal.
An inequality asks when one expression is less than, greater than, or possibly equal to another.
For example,
\[ x+3\lt 8 \]asks for every value of \(x\) that makes the left side smaller than \(8\).
Subtract \(3\) from both sides:
\[ x\lt5. \]The solution is not one number. It is every real number less than \(5\).
So solving an inequality means describing an entire solution set.
Inequality Symbols
| Symbol | Meaning |
|---|---|
| \(\lt\) | Less than |
| \(\gt\) | Greater than |
| \(\le\) | Less than or equal to |
| \(\ge\) | Greater than or equal to |
The strict symbols \(\lt\) and \(\gt\) exclude the boundary value.
The inclusive symbols \(\le\) and \(\ge\) include it.
One-Step Inequalities
Consider
\[ x+4\gt9. \]Subtract \(4\) from both sides:
\[ x\gt5. \]Every number greater than \(5\) satisfies the original inequality.
Addition and Subtraction Preserve Direction
Adding or subtracting the same quantity from both sides does not reverse the inequality.
For example,
\[ x-7\le2 \]becomes
\[ x\le9 \]after adding \(7\) to both sides.
Likewise, from
\[ x+6\ge1, \]subtract \(6\):
\[ x\ge-5. \]Multiplication by a Positive Number Preserves Direction
Consider
\[ \frac{x}{3}\lt4. \]Multiply both sides by the positive number \(3\):
\[ x\lt12. \]Because the multiplier is positive, the order direction remains unchanged.
The Crucial Rule: Negative Multiplication Reverses Order
Suppose
\[ 2\lt5. \]Multiply both sides by \(-1\):
\[ -2\gt-5. \]The direction reverses.
Therefore, whenever both sides of an inequality are multiplied or divided by a negative quantity, the inequality symbol must reverse:
\[ \lt\ \longleftrightarrow\ \gt \]and
\[ \le\ \longleftrightarrow\ \ge. \]Why the Sign Reverses
Negative multiplication reflects numbers across zero on the number line.
For example,
\[ 3\lt7. \]After multiplying by \(-1\):
\[ -3\gt-7. \]The point that was farther right becomes farther left after reflection.
The sign reversal is therefore not an arbitrary rule. It preserves the original ordering relationship after reflection.
One-Step Negative-Coefficient Example
Solve
\[ -4x\lt20. \]Divide both sides by \(-4\).
Because the divisor is negative, reverse the inequality:
\[ x\gt-5. \]This reversal is essential.
Another Negative Example
Solve
\[ -3x\ge12. \]Divide by \(-3\) and reverse the direction:
\[ x\le-4. \]Two-Step Inequalities
Solve
\[ 3x+5\lt20. \]Subtract \(5\):
\[ 3x\lt15. \]Divide by \(3\):
\[ x\lt5. \]This looks almost identical to solving a linear equation.
Two-Step Inequality with a Negative Coefficient
Solve
\[ -2x+3\gt11. \]Subtract \(3\):
\[ -2x\gt8. \]Divide by \(-2\) and reverse the inequality:
\[ x\lt-4. \]Multistep Inequalities
Consider
\[ 4x+7-2x\le19. \]Combine like terms:
\[ 2x+7\le19. \]Subtract \(7\):
\[ 2x\le12. \]Divide by \(2\):
\[ x\le6. \]Distribution in Inequalities
Solve
\[ 3(x-2)\gt12. \]We could divide by \(3\) first:
\[ x-2\gt4, \]then add \(2\):
\[ x\gt6. \]Or distribute first:
\[ 3x-6\gt12, \]then solve.
Both methods preserve the same solution set.
Negative Distribution
Consider
\[ -2(x+3)\le10. \]One approach is to divide by \(-2\) first.
Reverse the inequality:
\[ x+3\ge-5. \]Subtract \(3\):
\[ x\ge-8. \]This avoids distributing a negative coefficient and can reduce sign errors.
Variables on Both Sides
Solve
\[ 5x+2\gt3x+10. \]Subtract \(3x\):
\[ 2x+2\gt10. \]Subtract \(2\):
\[ 2x\gt8. \]Divide by \(2\):
\[ x\gt4. \]Choose the Variable Side Strategically
Consider
\[ 2x+7\lt5x-8. \]Subtract \(2x\):
\[ 7\lt3x-8. \]Add \(8\):
\[ 15\lt3x. \]Divide by \(3\):
\[ 5\lt x. \]More conventionally, write
\[ x\gt5. \]Choosing a route that keeps the variable coefficient positive can make sign handling easier.
Variables on Both Sides with a Negative Final Coefficient
Solve
\[ 7x+4\le3x-12. \]Subtract \(7x\):
\[ 4\le-4x-12. \]Add \(12\):
\[ 16\le-4x. \]Divide by \(-4\) and reverse the inequality:
\[ -4\ge x. \]So
\[ x\le-4. \]Fractions in Linear Inequalities
Solve
\[ \frac{x}{3}+\frac12\gt2. \]Subtract \(\frac12\):
\[ \frac{x}{3}\gt\frac32. \]Multiply by \(3\):
\[ x\gt\frac92. \]Clearing Positive Denominators
For
\[ \frac{x}{4}+\frac{x}{2}\le9, \]multiply every term by the positive number \(4\):
\[ x+2x\le36. \]Then
\[ 3x\le36, \]so
\[ x\le12. \]Because \(4\) is positive, the inequality direction stays unchanged.
Be Careful with Unknown-Sign Multipliers
Suppose
\[ ax\lt b. \]We cannot automatically divide by \(a\) without knowing its sign.
If
\[ a\gt0, \]then
\[ x\lt\frac{b}{a}. \]If
\[ a\lt0, \]then the inequality reverses:
\[ x\gt\frac{b}{a}. \]If
\[ a=0, \]the equation must be analyzed separately.
Parameter inequalities naturally require case analysis.
Solution Sets on the Number Line
The solution
\[ x\lt5 \]contains every real number to the left of \(5\).
The boundary value \(5\) is excluded.
For
\[ x\le5, \]the same leftward region is included, but now \(5\) belongs to the solution set.
Inequalities naturally describe regions rather than isolated points.
Open and Closed Boundaries
On a number line:
- \(x\lt a\) or \(x\gt a\) uses an open boundary because \(a\) is excluded.
- \(x\le a\) or \(x\ge a\) uses a closed boundary because \(a\) is included.
This same distinction appears in interval notation.
Interval Notation
The inequality
\[ x\lt5 \]can be written as
\[ (-\infty,5). \]The inequality
\[ x\le5 \]becomes
\[ (-\infty,5]. \]Similarly,
\[ x\gt2 \]becomes
\[ (2,\infty), \]and
\[ x\ge2 \]becomes
\[ [2,\infty). \]Why Infinity Always Uses Parentheses
The symbols
\[ \infty \]and
\[ -\infty \]do not represent ordinary endpoint numbers that can be included.
Therefore, intervals extending without bound always use parentheses at infinity:
\[ (-\infty,4] \]or
\[ (7,\infty). \]Testing a Solution
Suppose we solve
\[ 2x+1\lt9 \]and obtain
\[ x\lt4. \]Choose a value inside the solution set, such as \(x=0\):
\[ 2(0)+1=1, \]and
\[ 1\lt9. \]Choose a value outside, such as \(x=5\):
\[ 2(5)+1=11, \]and
\[ 11\not\lt9. \]Testing values can help catch algebraic mistakes.
Boundary Testing
For
\[ x\le4, \]the boundary value \(4\) should satisfy the original inequality.
For
\[ x\lt4, \]the boundary value should make the two sides equal but should not count as a solution.
Boundary testing is especially useful when checking strict versus inclusive symbols.
Inequalities That Become Identities
Consider
\[ 2(x+3)\le2x+6. \]Expand:
\[ 2x+6\le2x+6. \]Subtract \(2x\):
\[ 6\le6. \]This is always true.
Therefore, every real number is a solution:
\[ \mathbb{R}. \]Inequalities That Become Contradictions
Consider
\[ 3(x+2)\gt3x+8. \]Expand:
\[ 3x+6\gt3x+8. \]Subtract \(3x\):
\[ 6\gt8. \]This is false.
Therefore, the solution set is
\[ \varnothing. \]Strict Identity Cases
Consider
\[ 4x+5\lt4x+5. \]Subtract \(4x\):
\[ 5\lt5. \]This is never true.
So
\[ \varnothing \]is the solution set.
By contrast,
\[ 4x+5\le4x+5 \]is true for every real \(x\).
Three Possible Outcomes
A linear inequality may produce:
| Outcome | Example Final Form | Solution Set |
|---|---|---|
| A proper interval | \(x\lt4\) | \((-\infty,4)\) |
| All real numbers | \(3\le3\) | \(\mathbb{R}\) |
| No solution | \(7\lt2\) | \(\varnothing\) |
Equivalent Inequalities
Two inequalities are equivalent when they have exactly the same solution set.
For example,
\[ 2x+4\lt10 \]is equivalent to
\[ 2x\lt6, \]which is equivalent to
\[ x\lt3. \]We may write
\[ 2x+4\lt10 \quad\Longleftrightarrow\quad 2x\lt6 \quad\Longleftrightarrow\quad x\lt3. \]Negative Multiplication Still Preserves Equivalence When Reversed
Consider
\[ -2x\lt8. \]Dividing by \(-2\) and reversing the inequality gives
\[ x\gt-4. \]These are equivalent inequalities:
\[ -2x\lt8 \quad\Longleftrightarrow\quad x\gt-4. \]The sign reversal is what preserves equivalence.
Graphical Interpretation
Solving
\[ 2x+1\lt7 \]means finding where the graph
\[ y=2x+1 \]lies below the horizontal line
\[ y=7. \]The two graphs are equal when
\[ 2x+1=7, \]so
\[ x=3. \]For values less than \(3\), the line \(y=2x+1\) lies below \(7\).
Thus,
\[ x\lt3. \]Contextual Inequalities
Suppose a venue can hold at most \(250\) people.
If \(n\) is attendance, then
\[ n\le250. \]The inequality expresses a capacity constraint rather than a single target value.
Many real-world restrictions naturally produce inequalities:
- budgets;
- maximum weights;
- minimum scores;
- speed limits;
- temperature ranges;
- production capacities;
- error tolerances.
Application: Budget Constraint
Suppose tickets cost \$12 each and you have at most \$90.
If \(n\) is the number of tickets, then
\[ 12n\le90. \]Divide by \(12\):
\[ n\le7.5. \]But \(n\) counts whole tickets, so the contextual solution is
\[ n\le7 \]for nonnegative integers \(n\).
The algebraic solution must be interpreted within the domain of the model.
Application: Minimum Requirement
Suppose a course requires an average of at least \(70\).
If \(a\) is the average, then
\[ a\ge70. \]The phrase at least corresponds to \(\ge\).
The phrase at most corresponds to \(\le\).
Language and Inequality Symbols
| Phrase | Typical Symbol |
|---|---|
| Less than | \(\lt\) |
| More than / greater than | \(\gt\) |
| At most / no more than | \(\le\) |
| At least / no less than | \(\ge\) |
| Below | Usually \(\lt\) |
| Above | Usually \(\gt\) |
Worked Example: Multistep Inequality
Solve
\[ 5x-7\ge18. \]Add \(7\):
\[ 5x\ge25. \]Divide by \(5\):
\[ x\ge5. \]Interval notation:
\[ [5,\infty). \]Worked Example: Negative Coefficient
Solve
\[ -6x+4\lt22. \]Subtract \(4\):
\[ -6x\lt18. \]Divide by \(-6\) and reverse the inequality:
\[ x\gt-3. \]Interval notation:
\[ (-3,\infty). \]Worked Example: Variables on Both Sides
Solve
\[ 8x+3\le5x+18. \]Subtract \(5x\):
\[ 3x+3\le18. \]Subtract \(3\):
\[ 3x\le15. \]Divide by \(3\):
\[ x\le5. \]Interval notation:
\[ (-\infty,5]. \]Worked Example: Identity
Solve
\[ 4(x+2)-3\ge4x+5. \]Expand:
\[ 4x+8-3\ge4x+5. \]Simplify:
\[ 4x+5\ge4x+5. \]Subtract \(4x\):
\[ 5\ge5. \]This is always true.
Therefore,
\[ \boxed{\mathbb{R}}. \]Worked Example: Contradiction
Solve
\[ 2(x+4)\lt2x+3. \]Expand:
\[ 2x+8\lt2x+3. \]Subtract \(2x\):
\[ 8\lt3. \]This is false.
Therefore,
\[ \boxed{\varnothing}. \]A Reliable Solving Strategy
- Simplify each side.
- Collect variable terms on one side.
- Collect constants on the other side.
- Isolate the variable.
- Reverse the inequality whenever multiplying or dividing by a negative quantity.
- Interpret the result as a solution set.
- Write interval notation or graph the region when useful.
- Apply contextual restrictions such as integer counts or physical bounds.
Common Errors
Forgetting to Reverse the Inequality
From
\[ -2x\lt6, \]the correct result is
\[ x\gt-3. \]Reversing the Sign During Addition or Subtraction
Adding or subtracting the same quantity does not reverse the inequality.
Reversing Because a Negative Number Appears Somewhere
The symbol reverses only when the entire inequality is multiplied or divided by a negative quantity.
Using Brackets with Infinity
Infinity always uses parentheses in interval notation.
Ignoring Whether the Endpoint Is Included
The symbols \(\le\) and \(\ge\) include the boundary; \(\lt\) and \(\gt\) do not.
Forgetting Context
If a variable counts people, objects, or tickets, the real-number interval may need to be restricted to appropriate integers.
Practice
1. One-Step Addition
Solve:
\[ x+6\gt10. \]Check Answer
\[ x\gt4. \]Interval notation:
\[ (4,\infty). \]2. Positive Multiplication
Solve:
\[ 3x\le21. \]Check Answer
\[ x\le7. \]Interval notation:
\[ (-\infty,7]. \]3. Negative Multiplication
Solve:
\[ -5x\gt20. \]Check Answer
Divide by \(-5\) and reverse:
\[ x\lt-4. \]Interval notation:
\[ (-\infty,-4). \]4. Two-Step Inequality
Solve:
\[ 4x-3\ge13. \]Check Answer
\[ 4x\ge16, \]so
\[ x\ge4. \]5. Variables on Both Sides
Solve:
\[ 7x+2\lt4x+14. \]Check Answer
\[ 3x+2\lt14, \] \[ 3x\lt12, \] \[ x\lt4. \]6. Identity
Solve:
\[ 3(x+1)\le3x+3. \]Check Answer
The inequality simplifies to
\[ 3\le3, \]which is always true.
So the solution set is
\[ \mathbb{R}. \]7. Contradiction
Solve:
\[ 2x+5\gt2x+9. \]Check Answer
The inequality reduces to
\[ 5\gt9, \]which is false.
Therefore,
\[ \varnothing. \]Key Ideas
- Linear inequalities usually describe intervals or regions rather than single values.
- Addition and subtraction preserve the direction of an inequality.
- Multiplication or division by a positive quantity preserves the direction.
- Multiplication or division by a negative quantity reverses the inequality.
- The reversal rule reflects the order-reversing effect of multiplying by a negative number on the number line.
- Variables can appear on both sides and can be collected just as in equations.
- Strict inequalities exclude boundary values; inclusive inequalities include them.
- Solutions can be represented symbolically, with interval notation, or as regions on a number line.
- An inequality may be true for every value, false for every value, or true only on a proper subset of the domain.
- Unknown-sign symbolic divisors require case analysis.
- Testing interior, exterior, and boundary values can help verify a solution set.
- Real-world inequalities often represent limits, minimums, maximums, capacities, and allowable ranges.
Where This Leads
Linear inequalities extend equation solving from individual solution values to ordered regions of the number line.
This prepares the way for compound inequalities, absolute-value inequalities, polynomial and rational inequalities, systems of inequalities, feasible regions, optimization, and the broader study of mathematical constraints.