CSCA MATHEMATICS · MODULE 01 · LESSON 01.4
Algebraic Inequalities
Master inequality properties, solve linear and compound inequalities, convert solutions into intervals, and avoid common exam mistakes.
Learning objectives
- Interpret strict and inclusive inequalities.
- Apply addition, subtraction, multiplication and division rules safely.
- Solve linear inequalities, including expressions with parentheses or fractions.
- Identify all-real and no-solution cases.
- Represent AND/OR solutions using intervals and number lines.
- Translate word problems into inequalities and check answers.
01.4.1 Understanding Algebraic Inequalities
An inequality compares two quantities. It may use fixed numbers or variables. For example:
The first means 5 is greater than 3; the second means 2 is less than 7.
Worked Example 1 · Solve a simple inequality
Start with the inequality:
Subtract 3 from both sides:
Final answer
Values such as 6, 7 and 10 work; 5 does not because \(5+3=8\) is not greater than 8.
The four main inequality symbols
| Symbol | Meaning | Example |
|---|---|---|
| > | Greater than | \(x>4\) |
| < | Less than | \(x<4\) |
| ≥ | Greater than or equal to | \(x\geq4\) |
| ≤ | Less than or equal to | \(x\leq4\) |
A strict inequality excludes its endpoint. A non-strict inequality includes it. Thus \(x>4\) excludes 4, but \(x\geq4\) includes 4.
Equations versus inequalities
| Equation | Inequality |
|---|---|
| \(x+2=7\) | \(x+2<7\) |
| \(x=5\): one solution | \(x<5\): infinitely many real solutions |
Key concept
Solving an inequality means finding every value that makes the statement true.
01.4.2 Fundamental Properties of Inequalities
Property 1 · Addition
Adding the same real number to both sides preserves the direction of an inequality.
Worked Example 2 · Addition examples
Add 5 to both sides:
Result:
Solve:
Add 6:
Final answer
Interval: \((-\infty,16)\)
Property 2 · Subtraction
Subtracting the same real number from both sides also preserves the inequality direction.
Worked Example 3 · Subtraction examples
Subtract 4:
Result:
Solve:
Subtract 9:
Final answer
Interval: \([6,\infty)\)
Property 3 · Multiplication
Multiplying by a positive number preserves the sign. Multiplying by a negative number reverses it.
| Positive multiplier | Negative multiplier |
|---|---|
| \(2<5\) | \(2<5\) |
| \(2(3)<5(3)\) | \(2(-3)>5(-3)\) |
| \(6<15\) — unchanged | \(-6>-15\) — reversed |
Essential exam rule
When you multiply or divide an inequality by a negative quantity, reverse the inequality sign.
Property 4 · Division
Division by a positive nonzero number preserves the direction; division by a negative number reverses it.
Worked Example 4 · Division by a positive number
Given:
Divide by 4:
Final answer
Worked Example 5 · Division by a negative number
Given:
Divide by −3 and reverse the sign:
Final answer
Summary of arithmetic properties
| Operation on both sides | Direction |
|---|---|
| Add any real number | Unchanged |
| Subtract any real number | Unchanged |
| Multiply by a positive number | Unchanged |
| Multiply by a negative number | Reversed |
| Divide by a positive number | Unchanged |
| Divide by a negative number | Reversed |
Zero and variable multipliers
Never divide by zero. Multiplying by zero does not give an equivalent inequality. If multiplying or dividing by a variable expression, establish whether it is positive, negative, or zero first.
01.4.3 Additional Properties of Inequalities
Property 5 · Transitivity
If a quantity is smaller than a second quantity, which is smaller than a third, then the first is smaller than the third.
Worked Example 6 · Transitivity
Given:
Conclude:
Mixed case:
Property 6 · Reversing the order
Swapping the two sides also reverses the direction symbol.
Property 7 · Negation
Multiplying both sides by −1 reverses the sign.
01.4.4 Solving Basic Linear Inequalities
A linear inequality in one variable can have forms such as:
Here \(a,b,c\) are real constants with \(a
eq0\). The goal is to isolate the variable.
General solving procedure
- Simplify brackets and combine like terms.
- Collect variable terms on one side.
- Collect constants on the other side.
- Divide by the coefficient of the variable.
- Reverse the sign if the divisor is negative.
- Write the solution in inequality and interval notation.
- Check a value and the boundary.
Worked Example 11 · Two-step inequality
Start:
Add 7:
Divide by 5:
Final answer
Interval: \((-\infty,5]\)
Worked Example 12 · Negative coefficient
Start:
Subtract 9:
Divide by −4; reverse the sign:
Final answer
Interval: \((-\infty,-3)\)
Verify using \(x=-4\) (inside the solution) and \(x=-2\) (outside):
Worked Example 13 · Variables on both sides
Start:
Subtract 2x:
Subtract 5:
Final answer
Interval: \((6,\infty)\)
Why moving terms works
Moving a term across an inequality is a shortcut for adding or subtracting the same term on both sides.
01.4.5 Solving Inequalities with Parentheses
Use the distributive property before collecting like terms:
Worked Example 14 · Expanding parentheses
Start:
Expand:
Subtract 4x:
Add 2:
Divide by 2:
Final answer
Interval: \((-\infty,4]\)
Worked Example 15 · Negative coefficient outside parentheses
Start:
Distribute −2:
Add 6:
Divide by −2; reverse:
Final answer
Interval: \((-\infty,-7]\)
Common mistake
Distribute the negative coefficient to every term: \(-2(x+3)=-2x-6\), not \(-2x+6\).
01.4.6 Solving Inequalities with Fractions
If denominators are positive constants, multiply both sides by their positive least common multiple (LCM). The inequality direction is unchanged.
Worked Example 16 · Clearing denominators
Start:
LCM of 4 and 3:
\mathrm{LCM}(4,3)=12
\]
Multiply both sides by 12:
Simplify:
Expand:
Subtract 3x:
Subtract 8:
Final answer
Interval: \([-11,\infty)\)
Important limitation
If a denominator contains a variable, you must check where it equals zero and analyze its sign. Simply clearing denominators may be invalid.
01.4.7 All Real Numbers or No Solution
If the variable disappears during simplification, the remaining numerical comparison determines whether all real numbers or none are solutions.
Case 1 · All real numbers
Consider:
Expand and simplify:
This is always true, so every real number is a solution.
x ∈ ℝ
Interval: \((-\infty,\infty)\)
Case 2 · No solution
Consider:
Expand and simplify:
This is false for every x, so there is no real solution.
∅
| Simplified statement | Meaning | Solution set |
|---|---|---|
| Always true, e.g. 5 < 8 | All real numbers satisfy the original inequality | ℝ |
| Always false, e.g. 9 < 4 | No real number satisfies the original inequality | ∅ |
01.4.8 Basic Compound Inequalities
A compound inequality combines conditions using AND or OR. The difference matters: AND requires both conditions, while OR requires at least one.
Type 1 · AND
Worked Example A · Intersection of conditions
Start:
Add 1 to all three parts:
Divide every part by 3:
Final answer
Interval: \([1,4)\)
The result includes 1 but excludes 4.
Type 2 · OR
Worked Example B · Union of separate intervals
First inequality:
Solve the first:
Second inequality:
Solve the second:
Combine using OR:
Final answer
AND vs OR
AND means shared solutions (intersection). OR means solutions satisfying at least one condition (union).
01.4.9 Representing Solutions on a Number Line
Number lines show which values satisfy an inequality. An open circle excludes the boundary; a filled circle includes it.
Example A · x > 2
0
1
2
3
4
Open circle at 2, highlight to the right. Interval: (2, ∞).
Example B · x ≤ −1
−3
−2
−1
0
1
Filled circle at −1, highlight to the left. Interval: (−∞, −1].
Interval notation summary
| Inequality | Interval notation |
|---|---|
| \(x>a\) | \((a,\infty)\) |
| \(x\geq a\) | \([a,\infty)\) |
| \(x<a\) | \((-\infty,a)\) |
| \(x\leq a\) | \((-\infty,a]\) |
| \(a<x<b\) | \((a,b)\) |
| \(a\leq x\leq b\) | \([a,b]\) |
Remember
Infinity is never included as an endpoint; use a parenthesis next to ∞ or −∞.
01.4.10 Applying Inequalities to Word Problems
Inequalities describe limits, minimum requirements, and allowable ranges.
| Phrase | Mathematical meaning |
|---|---|
| More than | > |
| Less than | < |
| At least | ≥ |
| At most | ≤ |
| No more than | ≤ |
| No less than | ≥ |
| Exceeds | > |
Worked Example 17 · Budget problem
A textbook costs $10, each workbook costs $4, and the total budget is $38. Let n be the number of workbooks:
Subtract 10:
Divide by 4:
Because n is a nonnegative integer:
Final answer
Interpretation: The student can buy at most seven practice workbooks.
01.4.11 Common Mistakes to Avoid
Mistake 1 · Forgetting to reverse the sign
| Incorrect | Correct |
|---|---|
| \(-2x>8\) | \(-2x>8\) |
| \(x>-4\) | \(x<-4\) |
Dividing by −2 reverses the inequality.
Mistake 2 · Treating strict inequalities as inclusive
The condition \(x<5\) excludes 5. It is not the same as \(x\leq5\).
Mistake 3 · Incorrect distribution
The negative coefficient multiplies both terms inside the parentheses.
Mistake 4 · Multiplying by a variable of unknown sign
Do not multiply by x and assume the direction stays the same; x may be positive or negative, and x = 0 is excluded.
Mistake 5 · Giving just one solution
For \(x>3\), a single value such as 4 is only an example. The complete solution contains infinitely many real numbers.
Mistake 6 · Forgetting real-world constraints
Counts of books, people, or questions must generally be nonnegative integers, even if algebraic operations initially produce a real-number interval.
01.4.12 Lesson Summary
Review these principles before attempting the topic quiz:
- Adding and subtracting the same quantity preserves the inequality direction.
- Multiplying or dividing by a positive number preserves the direction.
- Multiplying or dividing by a negative number reverses the direction.
- To solve a linear inequality, simplify and isolate the variable.
- AND means intersection; OR means union.
- An always-true simplified statement gives all real solutions; an always-false one gives none.
- Use intervals and number lines to display solution sets.
Essential sign rule
For nonzero c and \(a<b\):
The same principle applies to division and to inclusive inequalities.
Completion target
Solve basic linear and compound inequalities, verify boundary inclusion, explain sign reversals, and express solutions in interval notation.