Compound Inequalities

A compound inequality combines two inequalities to describe a solution set with more than one condition.

For example:

x > 2 and x < 7

means that x must satisfy both conditions.

The solution contains numbers greater than 2 but less than 7:

2 < x < 7

Another example is:

x < −3 or x > 4

Here, x only needs to satisfy one of the two conditions.

Understanding the difference between and and or is the key to working with compound inequalities.

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What Makes an Inequality Compound?

A simple inequality contains one comparison:

x > 5

A compound inequality combines two comparisons:

x > 5 and x < 10

or:

x < −2 or x ≥ 6

Compound inequalities are useful whenever a quantity has:

• both a lower and upper limit
• two separate acceptable ranges
• multiple conditions that must be satisfied

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The Meaning of "And"

An and inequality requires both conditions to be true at the same time.

Consider:

x > 2 and x < 8

For a number to be a solution, it must be:

• greater than 2
• less than 8

Therefore:

2 < x < 8

Examples of solutions include:

3, 4, 5, 6, 7, 2.5, and 7.9.

But 2 and 8 are not included.

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"And" Means Intersection

The solution to an and inequality is the overlap of the two individual solution sets.

For:

x > 2 and x < 8

the first condition gives numbers to the right of 2.

The second gives numbers to the left of 8.

The overlap is:

2 < x < 8

This overlap is called the intersection.

A useful way to remember this is:

AND = OVERLAP

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Writing an "And" Inequality as One Statement

The statement:

x ≥ 3 and x < 9

can be written:

3 ≤ x < 9

Read this as:

"x is greater than or equal to 3 and less than 9."

The boundary 3 is included.

The boundary 9 is not included.

Interval notation:

[3, 9)

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Double Inequalities

Expressions such as:

2 < x < 10

are called double inequalities.

The expression really contains two statements:

2 < x

and:

x < 10

Together:

x > 2 and x < 10

The variable lies between two boundaries.

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Reading Double Inequalities

Consider:

−4 ≤ x < 6

This means:

x ≥ −4 and x < 6

Therefore:

• −4 is included
• 6 is not included
• every number between them is included

Interval notation:

[−4, 6)

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The Meaning of "Or"

An or inequality requires at least one of the conditions to be true.

Consider:

x < −2 or x > 5

The solution includes:

• all numbers less than −2
• all numbers greater than 5

Examples include:

−10, −3, 6, 20, and 100.

Numbers between −2 and 5 do not satisfy either condition.

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"Or" Means Union

For:

x < −2 or x > 5

the solution combines two separate regions.

This is called the union of the solution sets.

A useful way to remember this is:

OR = COMBINE

"And" usually looks for overlap.

"Or" usually combines the acceptable regions.

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Comparing "And" and "Or"

Consider these two statements:

x > 2 and x < 7

Solution:

2 < x < 7

Now:

x < 2 or x > 7

Solution:

all numbers outside the interval from 2 to 7.

Graphically, these have very different appearances.

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Testing an "And" Statement

Consider:

x ≥ 4 and x < 10

Test:

x = 6

First condition:

6 ≥ 4

True.

Second condition:

6 < 10

True.

Both are true, so:

x = 6 is a solution.

Now test x = 12.

12 ≥ 4 is true.

12 < 10 is false.

Because both conditions must be true:

x = 12 is not a solution.

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Testing an "Or" Statement

Consider:

x < −3 or x ≥ 5

Test:

x = 8

First condition:

8 < −3

False.

Second condition:

8 ≥ 5

True.

Because at least one condition is true:

x = 8 is a solution.

For an "or" statement, both conditions do not need to be true.

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Number Lines for "And" Inequalities

Consider:

−2 < x ≤ 5

The number-line representation has:

• open circle at −2
• closed circle at 5
• shading between the two boundaries

The open circle means −2 is excluded.

The closed circle means 5 is included.

Interval notation:

(−2, 5]

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Number Lines for "Or" Inequalities

Consider:

x ≤ −2 or x > 5

The graph has:

• closed circle at −2 and shading left
• open circle at 5 and shading right

There are two separate regions.

Interval notation:

(−∞, −2] ∪ (5, ∞)

The symbol ∪ means union.

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Interval Notation Review

Interval notation uses parentheses and square brackets.

means the endpoint is not included.

[ ] means the endpoint is included.

For example:

2 < x ≤ 8

becomes:

(2, 8]

The 2 uses a parenthesis because it is excluded.

The 8 uses a bracket because it is included.

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Infinity in Interval Notation

Infinity always uses parentheses.

For:

x < 4

interval notation is:

(−∞, 4)

For:

x ≥ 4

interval notation is:

[4, ∞)

We never write:

[∞]

because infinity is not an actual endpoint that can be included.

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Union Notation

When an "or" inequality produces two separate intervals, use the union symbol:

∪

For example:

x < −1 or x ≥ 6

becomes:

(−∞, −1) ∪ [6, ∞)

This means:

the first interval or the second interval.

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Solving an "And" Compound Inequality

Consider:

x + 3 > 5 and x + 3 < 11

Solve each condition.

First:

x + 3 > 5

x > 2

Second:

x + 3 < 11

x < 8

Combine:

2 < x < 8

Interval notation:

(2, 8)

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Solving a Double Inequality Together

A double inequality can often be solved as one expression.

Consider:

3 < x + 2 ≤ 9

Subtract 2 from all three parts:

3 − 2 < x + 2 − 2 ≤ 9 − 2

Therefore:

1 < x ≤ 7

Interval notation:

(1, 7]

This is often more efficient than separating the inequality into two statements.

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Performing Operations on All Three Parts

Consider:

−4 ≤ 2x < 10

Divide all three parts by 2:

−2 ≤ x < 5

Therefore:

−2 ≤ x < 5

Interval notation:

[−2, 5)

When solving a double inequality, whatever operation is performed must be applied throughout the entire inequality.

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Multiplying a Double Inequality by a Negative

The negative-number rule still applies.

Consider:

−8 < −2x ≤ 6

Divide every part by −2.

Because we are dividing by a negative number, both inequality symbols reverse:

4 > x ≥ −3

It is clearer to rewrite this from smallest to largest:

−3 ≤ x < 4

Interval notation:

[−3, 4)

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Why Both Symbols Reverse

Start with:

−8 < −2x ≤ 6

After division by −2:

−8/−2 > x ≥ 6/−2

giving:

4 > x ≥ −3

Reordering from least to greatest gives:

−3 ≤ x < 4

Every comparison reverses because every part of the inequality was divided by a negative number.

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Solving a Multi-Step Double Inequality

Solve:

5 ≤ 2x + 1 < 13

Subtract 1 from all three parts:

4 ≤ 2x < 12

Divide all parts by 2:

2 ≤ x < 6

Interval notation:

[2, 6)

Number line:

• closed at 2
• open at 6
• shade between them

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Another Multi-Step Example

Solve:

−7 < 3x + 2 ≤ 14

Subtract 2:

−9 < 3x ≤ 12

Divide by 3:

−3 < x ≤ 4

Interval notation:

(−3, 4]

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Solving "Or" Inequalities Separately

"Or" inequalities are normally solved as two separate inequalities.

Consider:

2x < −6 or 3x ≥ 12

Solve the first:

2x < −6

x < −3

Solve the second:

3x ≥ 12

x ≥ 4

Therefore:

x < −3 or x ≥ 4

Interval notation:

(−∞, −3) ∪ [4, ∞)

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"Or" Example with Negative Coefficients

Solve:

−3x > 9 or 2x ≥ 10

First inequality:

−3x > 9

Divide by −3 and reverse:

x < −3

Second inequality:

2x ≥ 10

x ≥ 5

Therefore:

x < −3 or x ≥ 5

Interval notation:

(−∞, −3) ∪ [5, ∞)

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Compound Inequalities with Variables on Both Sides

Consider:

3x + 2 > x + 6 and 2x − 1 ≤ 9

Solve the first:

3x + 2 > x + 6

2x + 2 > 6

2x > 4

x > 2

Solve the second:

2x − 1 ≤ 9

2x ≤ 10

x ≤ 5

Both conditions must be true:

2 < x ≤ 5

Interval notation:

(2, 5]

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Finding the Overlap

Suppose we solve an "and" inequality and obtain:

x ≥ −1

and:

x < 4

The overlap is:

−1 ≤ x < 4

But suppose we obtain:

x > 6

and:

x < 2

No number can satisfy both conditions.

Therefore:

no solution

or:

∅

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"And" with No Solution

Consider:

x + 2 > 10 and x − 3 < 2

First:

x > 8

Second:

x < 5

The compound condition requires:

x > 8 and x < 5

No number satisfies both.

Therefore:

∅

The number line has no shaded region.

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Overlapping "Or" Solutions

Consider:

x > 2 or x > 6

Every number greater than 6 is already included in:

x > 2

Therefore, the combined solution is simply:

x > 2

Interval notation:

(2, ∞)

We do not need to write two intervals when one completely contains the other.

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An "Or" Inequality Giving All Real Numbers

Consider:

x < 5 or x > 2

Think about the two solution sets.

The first contains everything below 5.

The second contains everything above 2.

Together, they cover every real number.

Therefore:

all real numbers

Interval notation:

(−∞, ∞)

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Visualizing Overlap

Number lines are especially useful for compound inequalities.

For an "and" problem:

1. graph each condition
2. identify where the graphs overlap
3. keep only the overlap

For an "or" problem:

1. graph each condition
2. combine every region covered by either graph

This makes the logical difference between "and" and "or" visible.

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Practical Situation: Temperature Range

A laboratory sample must remain between:

2°C and 8°C, inclusive.

Let T represent temperature.

Because both boundaries are included:

2 ≤ T ≤ 8

Interval notation:

[2, 8]

This is an and situation because the temperature must satisfy both the minimum and maximum conditions.

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Practical Situation: Safe Operating Temperature

Suppose a machine operates safely only when its temperature is:

• at least 20°C
• below 70°C

Then:

20 ≤ T < 70

Interval notation:

[20, 70)

The machine must satisfy both conditions simultaneously.

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Practical Situation: Age Range

A program accepts students who are:

• at least 12 years old
• no older than 16

Let a represent age.

Then:

12 ≤ a ≤ 16

Interval notation:

[12, 16]

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Practical Situation: Values Outside a Range

Suppose a warning system activates when temperature is:

• below 0°C
• or above 100°C

Then:

T < 0 or T > 100

Interval notation:

(−∞, 0) ∪ (100, ∞)

This is an "or" condition because either extreme activates the warning.

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Practical Situation: Manufacturing Tolerance

A component should have a length of at least:

49.8 mm

and at most:

50.2 mm

Let L represent length.

Then:

49.8 ≤ L ≤ 50.2

Interval notation:

[49.8, 50.2]

This type of compound inequality is extremely important in manufacturing because real components must remain within acceptable tolerances.

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Practical Situation: Speed Range

Suppose a machine operates correctly only between:

1200 rpm and 1800 rpm

including both endpoints.

Let n represent rotational speed.

Then:

1200 ≤ n ≤ 1800

The acceptable interval is:

[1200, 1800]

A speed of 1500 rpm is acceptable.

A speed of 2000 rpm is not.

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Building a Compound Inequality from Words

Consider:

"A number is greater than −4 but no more than 10."

"Greater than −4" means:

x > −4

"No more than 10" means:

x ≤ 10

Because both conditions must hold:

−4 < x ≤ 10

Interval notation:

(−4, 10]

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Understanding "Between"

The word between can sometimes be ambiguous in everyday language.

For example:

"x is between 3 and 8"

might mean:

3 < x < 8

But if the endpoints are included, the wording should make this clear:

"x is between 3 and 8, inclusive"

means:

3 ≤ x ≤ 8

Always check whether the boundary values are allowed.

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Understanding "Outside"

Suppose:

"x lies outside the interval from 2 to 7."

If the boundaries are also outside the acceptable region, this could be written:

x < 2 or x > 7

If the boundaries are included in the outside region:

x ≤ 2 or x ≥ 7

The exact inequality depends on whether the boundary values belong to the solution set.

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Solving a Practical Multi-Step Problem

A freezer should operate between:

−20°C and −12°C

inclusive.

A sensor reports a transformed value:

T = 2x − 4

We require:

−20 ≤ 2x − 4 ≤ −12

Add 4:

−16 ≤ 2x ≤ −8

Divide by 2:

−8 ≤ x ≤ −4

Interval notation:

[−8, −4]

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Solving a Practical "Or" Problem

A machine automatically shuts down when a sensor value is:

less than −10 or greater than 15

Suppose the sensor value is:

2x + 3

Then:

2x + 3 < −10 or 2x + 3 > 15

Solve the first:

2x < −13

x < −6.5

Solve the second:

2x > 12

x > 6

Therefore:

x < −6.5 or x > 6

Interval notation:

(−∞, −6.5) ∪ (6, ∞)

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Verifying an "And" Solution

Suppose:

−2 ≤ x < 5

Choose:

x = 3

Check:

−2 ≤ 3

True.

3 < 5

True.

Both conditions are true.

Therefore, 3 is a solution.

Now test x = 6:

−2 ≤ 6

True.

6 < 5

False.

Therefore, 6 is not a solution.

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Verifying an "Or" Solution

Suppose:

x < −4 or x ≥ 7

Test x = 10:

10 < −4

False.

10 ≥ 7

True.

Because one condition is true, 10 is a solution.

Test x = 2:

2 < −4

False.

2 ≥ 7

False.

Neither condition is true, so 2 is not a solution.

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Checking Boundary Values

Boundary values are particularly important with compound inequalities.

Consider:

−3 < x ≤ 8

Test x = −3:

−3 < −3

False.

So −3 is excluded.

Test x = 8:

8 ≤ 8

True.

So 8 is included.

This agrees with:

(−3, 8]

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A Reliable Strategy for "And" Inequalities

When solving an "and" inequality:

1. Identify both conditions.
2. Solve each inequality.
3. Reverse inequality signs if multiplying or dividing by negatives.
4. Find the values satisfying both conditions.
5. Identify the overlap.
6. Write the final inequality.
7. Convert to interval notation.
8. Graph the solution.
9. Check important values, especially boundaries.

Remember:

AND = OVERLAP

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A Reliable Strategy for "Or" Inequalities

When solving an "or" inequality:

1. Separate the two inequalities.
2. Solve each one independently.
3. Reverse signs when required.
4. Keep values satisfying either condition.
5. Combine the solution sets.
6. Use ∪ when separate intervals remain.
7. Graph both regions.
8. Check boundary values.

Remember:

OR = COMBINE

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Common Mistakes

Confusing "and" with "or"

"And" requires both conditions. "Or" requires at least one.

Graphing an "and" solution as two outward rays

"And" frequently produces the region between two boundaries.

Graphing an "or" solution only between the boundaries

"Or" often produces two separate regions outside a central interval.

Forgetting to reverse inequality signs

If you multiply or divide by a negative number, reverse the inequality.

Reversing only one symbol in a double inequality

If every part is divided or multiplied by a negative number, both inequality symbols reverse.

Forgetting to operate on all three parts

For:

2 < x + 5 ≤ 10

subtract 5 from all three parts.

Using the wrong interval symbol

Use parentheses for excluded finite boundaries and brackets for included finite boundaries.

Using brackets with infinity

Infinity always uses parentheses.

Forgetting the union symbol

Separate "or" intervals are joined using:

∪

Failing to simplify overlapping intervals

For:

x > 2 or x > 5

the final solution is simply:

x > 2

Assuming every "and" inequality has a solution

Conditions can contradict each other.

Assuming every "or" inequality produces two intervals

Overlapping "or" conditions can combine into one interval or even all real numbers.

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Key Terms

Compound inequality — two inequalities combined to describe a solution set involving multiple conditions.

And inequality — compound inequality requiring both conditions to be satisfied.

Or inequality — compound inequality requiring at least one condition to be satisfied.

Double inequality — compact notation placing a variable between two inequality symbols, such as 2 < x ≤ 8.

Intersection — values shared by two solution sets; associated with "and."

Union — combined values belonging to either solution set; associated with "or."

Solution set — all values satisfying the compound inequality.

Boundary value — endpoint separating included and excluded values.

Interval notation — notation using parentheses and brackets to describe intervals of real numbers.

Union symbol (∪) — symbol used to combine separate intervals.

Open endpoint — boundary value not included in the solution.

Closed endpoint — boundary value included in the solution.

All real numbers — every real-number value, represented by (−∞, ∞).

Empty set (∅) — solution set containing no values.

Constraint — condition restricting the possible values of a variable.

Tolerance — acceptable range of variation around a desired measurement or operating value.

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Key Takeaways

• Compound inequalities combine two mathematical conditions.
• And means both conditions must be true.
• Or means at least one condition must be true.
• "And" solutions are found by identifying the intersection or overlap.
• "Or" solutions are found by taking the union of acceptable values.
• Double inequalities such as 2 < x ≤ 8 represent "and" conditions.
• Operations on a double inequality must be performed on all three parts.
• Multiplying or dividing by a negative reverses every affected inequality symbol.
• Number lines provide a useful visual method for identifying compound solution sets.
• Open circles correspond to excluded endpoints.
• Closed circles correspond to included endpoints.
• Parentheses indicate excluded finite endpoints in interval notation.
• Brackets indicate included finite endpoints.
• Infinity always uses parentheses.
• Separate "or" intervals are joined with the union symbol ∪.
• An "and" inequality can sometimes have no solution.
• An "or" inequality can sometimes simplify to one interval or all real numbers.
• Compound inequalities are useful for expressing ranges, tolerances, safety limits, operating conditions, and restrictions.
• Testing boundary values helps determine whether endpoints should be included.
• A complete solution should clearly communicate the inequality, interval notation, and graphical representation.

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Check Your Understanding

1. Explain the difference between an and compound inequality and an or compound inequality.
2. Write x > 3 and x < 9 as a double inequality and in interval notation.
3. Write x ≤ −4 or x > 6 in interval notation.
4. Solve: 2 < x + 4 ≤ 11. Give the solution as an inequality and in interval notation.
5. Solve: −12 ≤ 3x < 15. Describe the corresponding number-line graph.
6. Solve: −8 < −2x ≤ 10. Be careful when dividing by the negative coefficient.
7. Solve: 3x < −9 or 2x ≥ 8. Give the answer in interval notation.
8. Solve: 2x + 5 > 9 and 3x − 1 ≤ 14. Find the overlap of the two solution sets.
9. Solve: x > 8 and x < 3. Explain why the solution set is empty.
10. Simplify the solution: x ≥ 2 or x > 7. Explain why two intervals are unnecessary.
11. Determine whether x = 4 is a solution of −2 ≤ x < 4. Explain your answer using both the inequality and its interval notation.
12. A machine must operate at a temperature of at least 25°C but below 80°C. Write the compound inequality and interval notation.
13. A warning activates when pressure is below 30 kPa or above 90 kPa. Define a variable, write the compound inequality, and describe its number-line graph.
14. A manufactured component is acceptable when its length is between 74.95 mm and 75.05 mm, inclusive. Write the compound inequality and interval notation. Explain what the boundaries mean in the context of quality control.
15. A sensor is considered safe when −15 ≤ 3x + 6 ≤ 24. Solve the compound inequality, write the answer in interval notation, describe the number-line graph, and verify both boundary values.