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Compound Inequality — AND, OR, Solving & Examples

#Algebra

TL;DR

A compound inequality joins two inequalities with the word "and" or "or", "and" gives the intersection (values satisfying both), "or" gives the union (values satisfying either). This article shows how to solve and graph each type on a number line, writes the answers in interval notation, and flags where the AND/OR logic most often goes wrong.

How Do You Solve a Compound Inequality?

The method splits cleanly by connector.

• "And" inequalities, take the intersection. Solve each piece, then keep only the overlap. For the three-part form a<x<b, you can isolate x in the middle by doing the same operation to all three parts at once.

• "Or" inequalities, take the union. Solve each piece separately, then keep everything covered by either solution.

How do you tell whether a compound inequality is AND or OR?

Read the connector word, not the symbols. The literal word "and" signals intersection; the word "or" signals union. A common trap is assuming 2<x<5 and x<2 or x>5 are "opposites you solve the same way." They are not.

Solving "And" Compound Inequalities

For a<x<b, isolate x in the middle by applying each step to all three parts. Solve −1≤2x+3<7:

Subtract 3 from all three parts:

−4≤2x<4.

Divide all three parts by 2 (positive, signs stay):

−2≤x<2.

The solution is the band [-2,2).

Solving "Or" Compound Inequalities

Solve each inequality on its own, then union the results. Solve 3x−1<5 or 2x≥12:

First piece: 3x−1<5 implies x<2.

Second piece: 2x≥12 implies x≥6.

Union: x<2 or x≥6.

In interval notation: (-∞,2)∪[6,∞).

Examples of Compound Inequality

Example 1

Solve 1<x+4≤6 (an "and" inequality).

Subtract 4 from all three parts:

−3<x≤2.

Final answer: −3<x≤2, or (-3,2].

Example 2

Solve x−1≤3 or x+2>9.

First piece: x≤4.

Second piece: x>7.

Union: x≤4 or x>7.

Final answer: (-∞,4]∪(7,∞).

Example 3

A student solves −2<3x+1<10 improperly.

The fix: operate on all three parts together:

Subtract 1 from all parts: −3<3x<9.

Divide all three by 3: −1<x<3.

Final answer: −1<x<3.

Example 4

Solve the "and" inequality 5≤2x−3≤11.

Add 3 to all three parts:

8≤2x≤14.

Divide by 2:

4≤x≤7.

Graph: closed circles at 4 and 7.

Final answer: 4≤x≤7.

Example 5

Solve −4x+1≥9 or x−3>2.

First piece: x≤−2.

Second piece: x>5.

Union: x≤−2 or x>5.

Final answer: (-∞,−2]∪(5,∞).

Example 6

When does an "and" inequality have no solution?

Example: x>5 and x<2. No number satisfies both conditions.

Final answer: no solution; the solution set is ∅.

Why Two Conditions Beat One

• Bands are everywhere. Safe blood pressure, tolerances on machined parts, are often an "and".
• Exclusions are "or" inequalities. Each is a union of two rules.

Where Compound Inequalities Trip Students Up

Mistake 1: Swapping the logic of "and" and "or"

Correct way: "and" is the intersection; "or" is the union.

Mistake 2: Solving only one side of a three-part inequality

Correct way: do it to all three parts at once.

Mistake 3: Forgetting the sign flip

Correct way: every rule from a single inequality still applies.

Conclusion

• A compound inequality joins two inequalities with "and" or "or."
• _"And" gives the intersection, values satisfying both.
• _"Or" gives the union, values satisfying either.
• In the three-part form a<x<b, operate on all three parts at once, and a negative factor still flips the sign.
• An "and" with no overlap has no solution (∅).

Frequently Asked Questions

What is a compound inequality? A compound inequality is two inequalities joined by "and" or "or."

What is the difference between AND and OR? "And" gives the intersection; "or" gives the union.

How do you solve a three-part compound inequality? Apply each operation to all three parts simultaneously.

Can a compound inequality have no solution? An "and" inequality whose two pieces never overlap has an empty solution set.

Does the sign-flip rule apply to compound inequalities? Yes, flipping applies to each piece.