Compound inequalities — AND, OR — Maths notes

WHAT is a compound inequality?

HOW to read the logic:

Word	Logic	Set operation	Region on number line
AND	both true	intersection $\cap$	overlap of the two
OR	either true	union $\cup$	both pieces combined

WHY AND = intersection, OR = union (derive it, don't memorise)

Let $A = \{x : \text{condition 1 true}\}$ and $B = \{x : \text{condition 2 true}\}$ .

"condition 1 AND condition 2" is true exactly when $x$ is in $A$ and in $B$ . By the definition of intersection, that set is $A \cap B$ . ← Why this step? Intersection is literally defined as "in both."
"condition 1 OR condition 2" is true when $x$ is in $A$ or in $B$ (inclusive or). By the definition of union, that set is $A \cup B$ . ← Why this step? Union is literally defined as "in at least one."

So the set operation is not a rule to learn — it is the meaning of AND/OR.

Figure — Compound inequalities — AND, OR

The compact AND form $a < x < b$

HOW to solve $a < f(x) < b$ : apply the SAME operation to all three parts at once.

$a \le 2x - 1 \le 7$ Add $1$ to every part: $a{+}1 \le 2x \le 8 \Rightarrow 1 \le 2x \le 8$ (taking $a=0$ ). Divide every part by $2$ : $\tfrac12 \le x \le 4$ . Why this step? Whatever keeps the left inequality true must be done to the middle and right too, or the chain breaks.

Worked examples

Recall Feynman: explain to a 12-year-old

Imagine two spotlights on a dark stage. AND = "stand where BOTH lights hit you" — only the small patch where the beams cross is safe. OR = "stand where AT LEAST ONE light hits you" — much bigger, both patches count. Numbers on the line are like spots on the stage: AND keeps the overlap, OR keeps everything lit.

Active recall

AND joins two inequalities using which set operation?

Intersection (

\cap

) — the overlap where both are true.

OR joins two inequalities using which set operation?

Union (

\cup

) — everything covered by at least one.

Compact form

a<x<b

represents which connective?

AND (both

a<x

and

x<b

Can OR be written as

a<x<b

No; OR gives two disjoint pieces, so you must write two separate inequalities.

When solving

a<f(x)<b

, what must you do to each operation?

Apply it to all THREE parts simultaneously.

Dividing a three-part inequality by a negative number does what?

Flips BOTH relation signs at once.

Solve

x>4

AND

x<1

No solution (

\varnothing

) — rays don't overlap.

Solve

x<5

x>2

All reals

\mathbb{R}

— union covers the whole line.

Solve

x<-2

x\ge 3

(-\infty,-2)\cup[3,\infty)

Why is AND=intersection not a rule to memorise?

Because intersection is defined as "in both sets," which is exactly what AND means.

Connections

Linear inequalities in one variable — each half of a compound inequality is one of these.
Set operations — union and intersection — the logical backbone here.
Interval notation — how we write $(a,b)$ , $[a,b]$ , and unions.
Absolute value inequalities — $|x|<a$ becomes an AND; $|x|>a$ becomes an OR.
Number line representation — the visual tool for overlap vs union.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Compound inequality ka matlab hai do inequalities ko ek logical word se jodna — AND ya OR. Yaad rakhne ka sabse simple tareeka: AND matlab dono conditions ek saath true honi chahiye, isliye hum dono ke overlap (intersection) ko lete hain. OR matlab kam se kam ek condition true ho, isliye hum dono ko milaakar union lete hain. Yahi poora concept hai — baaki sab isi se nikalta hai.

Number line pe socho: AND ek chhota sa beech ka patch deta hai (jahan dono rays overlap karti hain), aur OR bada region deta hai (dono tukde saath). AND ko compact form $a<x<b$ mein likh sakte ho, par OR ko kabhi nahi — OR ke liye do alag inequalities likhni padti hain, jaise $x<-2$ ya $x\ge 3$ .

Solving karte waqt teen-part inequality $a<f(x)<b$ mein jo bhi operation karo, teeno parts pe ek saath karo. Aur jab negative number se divide ya multiply karo, to dono relation signs flip hote hain, sirf ek nahi — yeh sabse common galti hai. Ek aur trap: $x>4$ AND $x<1$ ka answer koi solution nahi (empty), lekin $x>4$ OR $x<1$ ka answer do pieces — dono opposite! Isliye pehle word (AND/OR) dekho, phir intersection ya union decide karo. Yeh skill real life constraints (temperature range, age discount) mein har jagah kaam aati hai.

Compound inequalities — AND, OR

WHAT is a compound inequality?

WHY AND = intersection, OR = union (derive it, don't memorise)

The compact AND form $a < x < b$

Worked examples

Active recall

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Algebra — Introduction & Intermediate

Connections