What is the fundamental difference between an equation and an inequality?

An equation states that two expressions are equal, typically yielding a finite number of specific solutions. An inequality states that two expressions are not equal (one is greater than, less than, etc.), typically yielding an infinite range of solutions.

When do I have to flip the inequality sign?

You must flip the sign only when you multiply or divide both sides of the inequality by a negative number. Adding, subtracting, multiplying by a positive, or dividing by a positive does NOT flip the sign.

How do I know whether to use "and" or "or" for compound inequalities?

If the variable is between two values (e.g., a b), it's an "or" situation. For absolute value, |expression| c is "or".

Why do absolute value inequalities split into two cases?

Because the absolute value function measures distance from zero. For |x| 5, x must be more than 5 units from zero (x 5). The two cases capture both the positive and negative directions.

Can an inequality have no solution?

Yes. For example, |x| < -3 has no solution because the absolute value of any number is always non-negative. Similarly, x² < -5 has no real solution because a squared real number is always non-negative.

MathematicsHigh School

Inequalities

Inequalities are mathematical statements that compare two expressions using symbols like <, >, ≤, and ≥. Unlike equations that seek a single answer, inequalities describe entire ranges of possible values.

This guide covers key definitions, linear and quadratic inequalities, absolute value inequalities, graphing on number lines, systems of inequalities, worked examples, common mistakes, and a practice quiz.

1Introduction: Beyond Equality

In mathematics, equations tell us when two expressions are exactly equal (e.g., x + 2 = 5). But many real-world situations involve comparisons where values are not necessarily equal. A speed limit is a maximum speed, not an exact one; a budget sets a maximum spending limit. This is where inequalities come in.

An inequality is a mathematical statement that compares two expressions using an inequality symbol. Instead of finding a single solution, we often find a range of solutions.

Picture This

A roller coaster requires riders to be at least 48 inches tall. If h represents height in inches, this rule is h ≥ 48 — an inequality. Any height from 48 inches and above satisfies the rule, giving us infinitely many solutions instead of just one.

Real-World Uses

Speed Limits

Your speed must be ≤ the posted limit. Driving at any speed up to 60 mph satisfies v ≤ 60.

Budgeting

Total spending must be ≤ your budget. If you have

00, then cost ≤ 200.

Temperature Ranges

Water is liquid when 0 < T < 100 (Celsius) — a compound inequality.

Grades & Scores

To pass, you need a score ≥ 60%. To get an A, you need score ≥ 90%.

2Key Definitions

Inequality

A mathematical statement comparing two expressions using <, >, ≤, or ≥.

Solution Set

All values that make the inequality true. Usually a range, not a single value.

Interval Notation

Parentheses ( ) for strict (<, >); brackets [ ] for inclusive (≤, ≥). E.g., x > 3 is (3, ∞).

Boundary Point

The value(s) that separate the number line into regions where the inequality is true or false.

Compound "And"

Both inequalities must be true. Written as a < x < b. Solution is the intersection.

Compound "Or"

At least one must be true. E.g., x < a OR x > b. Solution is the union.

The Four Inequality Symbols

less than

greater than

≤

less than or equal

≥

greater than or equal

3Linear Inequalities

Linear inequalities are solved just like linear equations, with one critical difference: multiplying or dividing by a negative number flips the inequality sign.

The Golden Rule

Multiply or divide by a negative → FLIP the sign

Example: Solve 3x - 5 ≥ 7

3x - 5 ≥ 7

3x ≥ 12 (add 5 to both sides)

x ≥ 4

Solution Set: [4, ∞)

Example with the Golden Rule: Solve -2x + 1 < 9

-2x + 1 < 9

-2x < 8 (subtract 1)

x > -4 (divide by -2, FLIP the sign!)

Solution Set: (-4, ∞)

Memory Tip

"Negative flip, don't let it slip!" — Whenever you multiply or divide both sides by a negative number, flip that inequality sign.

4Quadratic Inequalities

Quadratic inequalities involve expressions like ax² + bx + c. The approach: find the roots, divide the number line into intervals, and test each interval.

Solving Steps

Make one side zero: Rearrange so all terms are on one side.
Find boundary points: Set the expression equal to zero and solve (factor or use the quadratic formula).
Test intervals: Pick a test value from each interval and check if it satisfies the inequality.
Write the solution: Include boundary points if ≤ or ≥.

Example: Solve x² - x - 6 > 0

Factor: (x - 3)(x + 2) = 0 → roots at x = 3 and x = -2

Test x = -3: (-3)² - (-3) - 6 = 6 > 0 ✓

Test x = 0: (0)² - (0) - 6 = -6 > 0 ✗

Test x = 4: (4)² - (4) - 6 = 6 > 0 ✓

Solution Set: (-∞, -2) ∪ (3, ∞)

Parabola Opens Up (a > 0)

> 0: outside the roots

< 0: between the roots

Parabola Opens Down (a < 0)

> 0: between the roots

< 0: outside the roots

5Absolute Value Inequalities

The absolute value |x| measures the distance of x from zero. Absolute value inequalities split into two key cases.

"Less Than" (AND case)

|ax + b| < c

Becomes: -c < ax + b < c

Solution is between two values

"Greater Than" (OR case)

|ax + b| > c

Becomes: ax + b < -c OR ax + b > c

Solution is outside two values

Example 1: Solve |2x - 1| ≤ 5

-5 ≤ 2x - 1 ≤ 5

-4 ≤ 2x ≤ 6 (add 1 to all parts)

-2 ≤ x ≤ 3

Solution Set: [-2, 3]

Example 2: Solve |x + 3| > 4

x + 3 < -4 OR x + 3 > 4

x < -7 OR x > 1

Solution Set: (-∞, -7) ∪ (1, ∞)

Special Cases

|expression| < negative

No solution. Absolute value is always ≥ 0, so it can never be less than a negative number.

|expression| > negative

All real numbers. Absolute value is always ≥ 0, which is always greater than any negative number.

6Graphing Inequalities on a Number Line

Visualizing solution sets on a number line is crucial for understanding inequalities. The key is choosing the right circle type and shading direction.

Open Circle ○

For strict inequalities (< or >). The boundary point is NOT included.

Closed Circle ●

For inclusive inequalities (≤ or ≥). The boundary point IS included.

Shading Rules

For x > a or x ≥ a, shade to the right.
For x < a or x ≤ a, shade to the left.
For compound "and" (a < x < b), shade between a and b.
For compound "or" (x < a or x > b), shade outside a and b.

Interactive: Number Line Visualizer

Value (a):2

x ≥ 2Interval: [2, ∞)

7Systems of Inequalities

A system of inequalities consists of two or more inequalities solved simultaneously. For two variables, the solution is a region on the coordinate plane where all shaded areas overlap.

Graphing Steps

Graph each inequality: Replace with = to find the boundary line.
Choose line type: Solid for ≤ or ≥, dashed for < or >.
Test a point: Pick (0, 0) if it is not on the line. If true, shade that side.
Find the overlap: The solution is where all shaded regions intersect.

Example: y > x - 1 and y ≤ -2x + 5

y > x - 1

Boundary: y = x - 1 (dashed line)

Test (0, 0): 0 > -1 ✓

Shade above the line

y ≤ -2x + 5

Boundary: y = -2x + 5 (solid line)

Test (0, 0): 0 ≤ 5 ✓

Shade below the line

The solution region is where the two shaded areas overlap — the area above y = x - 1 and below y = -2x + 5.

Dashed Line

Used for < or >. Points on the line are NOT part of the solution.

Solid Line

Used for ≤ or ≥. Points on the line ARE part of the solution.

8Worked Examples

Example 1: Linear Inequality with Fractions

Solve: (2/3)x - 1/2 < 5/6

LCD = 6. Multiply all terms by 6:

4x - 3 < 5

4x < 8 (add 3)

x < 2

Solution Set: (-∞, 2)

Example 2: Quadratic Inequality

Solve: x² ≥ 4x + 12

x² - 4x - 12 ≥ 0 (make one side zero)

(x - 6)(x + 2) = 0 → x = 6, x = -2

Test x = -3: 9 + 12 - 12 = 9 ≥ 0 ✓

Test x = 0: -12 ≥ 0 ✗

Test x = 7: 49 - 28 - 12 = 9 ≥ 0 ✓

Solution Set: (-∞, -2] ∪ [6, ∞)

Example 3: Absolute Value (Less Than)

Solve: |5 - 2x| < 7

-7 < 5 - 2x < 7

-12 < -2x < 2 (subtract 5)

6 > x > -1 (divide by -2, FLIP!)

-1 < x < 6

Solution Set: (-1, 6)

Example 4: Absolute Value (Greater Than)

Solve: |3x + 6| - 2 ≥ 4

|3x + 6| ≥ 6 (isolate abs value: add 2)

3x + 6 ≤ -6 OR 3x + 6 ≥ 6

3x ≤ -12 → x ≤ -4

3x ≥ 0 → x ≥ 0

Solution Set: (-∞, -4] ∪ [0, ∞)

Example 5: Compound "And" Inequality

Solve: -1 ≤ (x + 3)/2 < 4

-2 ≤ x + 3 < 8 (multiply all parts by 2)

-5 ≤ x < 5 (subtract 3)

Solution Set: [-5, 5)

9Common Mistakes

Forgetting to Flip the Sign

The most common error. When dividing by a negative: -2x < 6 becomes x > -3, NOT x < -3.

Mixing "And" and "Or" for Absolute Values

|expression| < c is "and" (between). |expression| > c is "or" (outside). Don't swap them.

Wrong Circle Type on Number Lines

Open circle for < or > (not included). Closed circle for ≤ or ≥ (included). Also, ∞ always uses parentheses, never brackets.

Assuming Quadratic Roots Are the Solution

Finding the roots is only step one. You must test the intervals between and outside the roots to determine which regions satisfy the inequality.

Not Isolating Absolute Value First

Quick Revision Summary

✓Inequalities compare expressions using <, >, ≤, ≥ and produce a range of solutions.
✓The Golden Rule: Flip the inequality sign when multiplying or dividing by a negative.
✓Parentheses ( ) mean endpoints not included; brackets [ ] mean included.
✓∞ and -∞ always use parentheses, never brackets.
✓Linear: Solve like equations, apply the Golden Rule when needed.
✓Quadratic: Find roots → test intervals → determine solution regions.
✓Absolute value: |expr| < c → AND (between); |expr| > c → OR (outside).
✓Number lines: Open circle for strict, closed for inclusive. Shade the correct direction.
✓Systems (2D): Graph each inequality, solution is the overlapping shaded region.
✓Always check special cases for absolute value when the right side is zero or negative.

Frequently Asked Questions

What is the fundamental difference between an equation and an inequality?: An equation states that two expressions are equal, typically yielding a finite number of specific solutions. An inequality states that two expressions are not equal (one is greater than, less than, etc.), typically yielding an infinite range of solutions.
When do I have to flip the inequality sign?: You must flip the sign only when you multiply or divide both sides of the inequality by a negative number. Adding, subtracting, multiplying by a positive, or dividing by a positive does NOT flip the sign.
How do I know whether to use "and" or "or" for compound inequalities?: If the variable is between two values (e.g., a < x < b), it's an "and" situation. If the variable is outside two values (e.g., x < a or x > b), it's an "or" situation. For absolute value, |expression| < c is "and", and |expression| > c is "or".
Why do absolute value inequalities split into two cases?: Because the absolute value function measures distance from zero. For |x| < 5, x must be within 5 units of zero (-5 < x < 5). For |x| > 5, x must be more than 5 units from zero (x < -5 or x > 5). The two cases capture both the positive and negative directions.
Can an inequality have no solution?: Yes. For example, |x| < -3 has no solution because the absolute value of any number is always non-negative. Similarly, x² < -5 has no real solution because a squared real number is always non-negative.

Practice Quiz

Test your understanding of inequalities — select the correct answer for each question.

1.Solve: x + 4 > 7

2.Solve: -3x ≤ 9

3.Solve: |x| < 5

4.Solve: |x| > 3

5.Solve: x² ≤ 9

6.Solve: 2x + 1 > 7 AND x < 5

7.Solve: |2x - 1| ≥ 5

8.What is the boundary point for x > 4?

9.Solve: (x + 1)(x - 2) < 0

10.Graph the solution to x ≥ 2:

Final Study Advice

1.Always identify the type of inequality first (linear, quadratic, absolute value, or compound).
2.Draw a number line to visualize the solution set — it helps catch errors.
3.For quadratic inequalities, always test the intervals — do not assume the answer from the roots alone.
4.Remember: |expr| < c is AND, |expr| > c is OR. This pattern never changes.
5.Check your answer by substituting a value from your solution set back into the original inequality.