Integrated Math 2–3 Honors · Lesson Review · August 24, 2026

Compound Inequalities: AND, OR, and Solution Sets

Decide whether to use the overlap or the union, graph the result, write interval notation, and solve compound inequalities—including all-real and empty solution sets.

13-minute edited lesson9 chaptersCaptions + transcript

01 · Watch

Lesson video

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0:0001Lesson Overview 0:1602AND vs. OR 1:0803AND: Use the Overlap 3:2904Write AND as a Chain 4:4205OR: Use the Union 6:5206Solve an OR Inequality 8:0907Solve an AND Inequality 10:5608OR Can Give All Real Numbers 12:0109AND Can Give the Empty Set

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02 · Understand

Learning targets

Distinguish an AND intersection from an OR union.
Graph compound inequality solution sets on a number line.
Write compound solution sets in interval notation.
Solve AND and OR compound inequalities and recognize all-real or empty outcomes.

03 · Vocabulary

Key definitions

Use the set language to explain why a graph is an overlap, a union, all real numbers, or empty.

01Compound inequality: Two or more inequalities considered together as one solution-set question.
02AND compound inequality: A number is a solution only when it satisfies both inequalities—the intersection or overlap.
03OR compound inequality: A number is a solution when it satisfies either inequality—the union of the two sets.
04Intersection: The values shared by two sets. On a number line, this is where the graphs overlap.
05Union: Every value in either set. In interval notation, $\cup$ joins the intervals.
06Empty set: A set with no elements, written $\varnothing$ . It occurs when an AND compound has no overlap.

04 · Apply

Worked examples

Identify AND or OR before you combine the graphs.

AND overlap

$x\ge4\quad\text{and}\quad x<10$

Keep only the values that satisfy both inequalities.

4\le x<10\qquad[4,10)

OR union

$x\ge5\quad\text{or}\quad x<0$

Keep every value in either ray and join the intervals with union.

(-\infty,0)\cup[5,\infty)

Solve OR

$3x\le12\quad\text{or}\quad x+6\ge20$

$3x\le12\;\Longrightarrow\;x\le4$
$x+6\ge20\;\Longrightarrow\;x\ge14$
$(-\infty,4]\cup[14,\infty)$

Solve AND

$-4\le2x+6<10$

Subtract 6 from all three parts: $-10\le2x<4$ .
Divide all three parts by 2: $-5\le x<2$ .
Interval notation: $[-5,2)$ .

All real numbers

$2x\ge-6\quad\text{or}\quad x+10<20$

The two sets are $x\ge-3$ or $x<10$ . Together they cover every real number.

(-\infty,\infty)=\mathbb{R}

Empty set

$x+2\ge6\quad\text{and}\quad x\le2$

The two requirements become $x\ge4$ and $x\le2$ . No number can satisfy both.

\varnothing

05 · Avoid

Common mistakes

Using overlap for OR.

OR includes every value in either set, not just the shared region.

Using union for AND.

AND keeps only values that satisfy both conditions.

Closing a strict endpoint.

The inequality $x<10$ uses an open circle and a round parenthesis at 10.

Forgetting the union symbol.

Two separate OR intervals need ∪ between them.

Changing only the middle of a chain.

Apply each legal operation to all three parts.

Assuming every AND has a solution.

Disjoint regions give the empty set.

06 · Check yourself

Quick check

Decide first, then open the answer.

1Write

x>-2\quad\text{and}\quad x\le7

as a chain and in interval notation.

$-2<x\le7$ and $(-2,7]$ .

2Write

x<-1\quad\text{or}\quad x\ge3

in interval notation.

$(-\infty,-1)\cup[3,\infty)$ .

3Solve

-8\le3x+1<10

$-3\le x<3$ . Subtract 1 from all three parts, then divide all three parts by 3.

4What is the AND solution to

x<1

and

x\ge4

The empty set, $\varnothing$ . The two rays do not overlap.

5What is the OR solution to

x\le5

x>2

All real numbers, $(-\infty,\infty)$ . The union covers the entire number line.

07 · Revisit

Transcript and assigned practice

Search the corrected transcript for an example, then complete the eKadence worksheet Compound Inequalities.

Download corrected transcript