Compound Inequalities

By the end of this lesson, you will be able to solve and graph "and" inequalities as a single three-part statement, solve "or" inequalities as two separate ones, and say what each kind of solution set means in a real constraint.

Two Conditions at Once

Most real constraints come in pairs.

A reaction must run at $150°C$ or hotter, and at $200°C$ or cooler.

A driver is pulled over for going under $45$ mph, or over $65$ mph.

Those two sentences look similar and behave completely differently. The first needs both conditions true together. The second needs only one of them true.

Word	What it demands	The picture
and	Both at once	A segment between two points
or	At least one	Two rays pointing away from each other

Getting the word right is most of the work. Everything after it is inequality solving you already know.

"And": One Statement in Three Parts

An "and" inequality is written as a single sandwich:

$$-3 \leq 2x + 1 < 9$$

Read it as two claims about the same thing: $2x + 1$ is at least $-3$, and $2x + 1$ is below $9$.

To solve it, do everything to all three parts. The middle is where the variable lives, and the two outer parts have to keep up with it or the sandwich stops meaning what it meant.

"Or": Two Statements, Solved Separately

An "or" inequality is genuinely two problems:

$$2x + 1 < -5 \quad \text{or} \quad 3x - 2 \geq 7$$

Solve each one on its own, then put the two answers side by side. There is no sandwich to write, because the two solution sets do not touch. Between them sits a gap of numbers that satisfy neither.

The two pieces are joined with the union symbol $\cup$, which just means "and also these":

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

Example 1: An "And", Worked All the Way Through

Solve $-3 \leq 2x + 1 < 9$. Every move happens three times, once on each part.

Subtract 1 from all three parts -4 <= 2x < 8

Word

What it demands

The picture

and

Both at once

A segment between two points

At least one

Two rays pointing away from each other

Type	Written as	Solve by	Interval
And	$a < x < b$	Doing everything to all three parts	$[a, b]$, $(a, b)$, and so on
Or	Two inequalities	Solving each one separately	$(-\infty, a) \cup (b, \infty)$

Two Conditions at Once

"And": One Statement in Three Parts

"Or": Two Statements, Solved Separately

Example 1: An "And", Worked All the Way Through

Compound Inequalities

Two Conditions at Once

"And": One Statement in Three Parts

"Or": Two Statements, Solved Separately

Example 1: An "And", Worked All the Way Through

Worked Example 1: Your Turn

Example 2: An "Or", Worked All the Way Through

Worked Example 2: An "Or", Your Turn

Example 3: An "And" with a Negative in It

Worked Example 3: Negative Again, Your Turn

Worked Example 4: A Height Restriction

Example 4: The Same Situation, Both Ways Round

Common Mistakes

Where This Goes Next

Check Your Understanding

Summary