Compound Inequalities

Q: How do I tell whether a compound inequality is AND or OR?

If it's written as one sandwich like $-2 \le x < 4$, it's an AND — the value has to satisfy both ends at once. If the word 'or' appears between two separate inequalities, it's an OR. On a graph: one segment between two points is AND; two rays pointing away from each other is OR.

Q: Can an AND inequality have no solution?

Yes. If the two pieces don't overlap — like $x 5$ — no number satisfies both, so there's no solution. If you solve a sandwich and end with the larger number on the left, like $6 < x < 2$, that's the signal there's no overlap.

Q: Do I flip both signs when I divide a sandwich by a negative?

Yes — the flip rule applies to every inequality sign in the statement. After flipping, rewrite the sandwich so the smaller number sits on the left; $2 > x \ge -4$ and $-4 \le x < 2$ say exactly the same thing.

A compound inequality is two inequalities joined into one statement. There are only two kinds: AND, where a number has to satisfy both pieces at once, and OR, where satisfying either piece is enough. $-4 < x \le 1$ is an AND (it means $x > -4$ and $x \le 1$ at the same time), while $x < -3$ or $x \ge 2$ is an OR.

The two types look different on a number line, and that picture is the fastest way to keep them straight. AND solutions are one segment between two points — the overlap where both pieces agree. OR solutions are usually two rays heading in opposite directions. Once you can see the shape, the algebra is nothing new.

AND inequalities: the sandwich

AND compound inequalities are usually written in sandwich form, with the variable expression in the middle: $2 < x - 1 \le 6$ . That single line means both $x - 1 > 2$ and $x - 1 \le 6$ must be true.

The solving rule: whatever you do, do it to all three parts. To solve $2 < x - 1 \le 6$ , add $1$ to the left, the middle, and the right, giving $3 < x \le 7$ . The variable ends up alone in the middle, and the two outer numbers are the ends of your segment.

The flip rule still applies, to all three parts at once. If you divide the whole sandwich by a negative number, both inequality signs reverse — and the smaller number swaps to the other end.

OR inequalities: solve each piece separately

An OR inequality like $x + 5 < 2$ or $2x - 1 \ge 7$ is just two separate problems sharing a page. Solve each one on its own, then write the two answers with the word 'or' still between them: $x < -3$ or $x \ge 4$ .

Never merge an OR answer into a sandwich. $x < -3$ or $x \ge 4$ cannot be written as $4 \le x < -3$ — that sandwich claims a number is both at least $4$ and below $-3$ , which is impossible. OR answers stay as two pieces.

Reading and drawing the graphs

The endpoints follow the same circle rules as single inequalities: a filled circle means the endpoint is included ( $\le$ or $\ge$ ), an open circle means it isn't ( $<$ or $>$ ).

AND graphs shade the segment between the two circles. OR graphs shade outward: one ray to the left of one circle and one ray to the right of the other. If you're handed a graph and asked for the inequality, read the two circles first (open or filled), then check whether the shading is between them (AND) or away from them (OR).

The graph below shows the AND inequality $-4 < x \le 1$ : an open circle at $-4$ , a closed circle at $1$ , and shading across the whole segment between them.

Worked examples

Example 1: a sandwich with addition

Solve $2 < x - 1 \le 6$ .

Start with the compound inequality

2 < x - 1 \le 6

Add

1

to all three parts

3 < x \le 7

Check

x = 5

5 - 1 = 4

, and

2 < 4 \le 6

✓

Answer: $3 < x \le 7$

Example 2: a sandwich with two steps

Solve $-5 \le 3x + 1 < 13$ .

Start with the compound inequality

-5 \le 3x + 1 < 13

Subtract

1

from all three parts

-6 \le 3x < 12

Divide all three parts by

3

— positive, so no flip

-2 \le x < 4

Check

x = 0

3(0) + 1 = 1

, and

-5 \le 1 < 13

✓

Answer: $-2 \le x < 4$

Example 3: dividing a sandwich by a negative

Solve $-4 < -2x \le 8$ .

Start with the compound inequality

-4 < -2x \le 8

Divide all three parts by

-2

and flip both signs

2 > x \ge -4

Rewrite with the smaller number on the left

-4 \le x < 2

Check

x = 0

-2(0) = 0

, and

-4 < 0 \le 8

✓

Answer: $-4 \le x < 2$

Example 4: an OR inequality

Solve $x + 5 < 2$ or $2x - 1 \ge 7$ .

Solve the first piece: subtract

5

x < -3

Solve the second piece: add

1

, then divide by

2

x \ge 4

Keep the word 'or' between the answers

Check

x = -4

-4 + 5 = 1 < 2

✓ and

x = 4

2(4) - 1 = 7 \ge 7

✓

Answer: $x < -3$ or $x \ge 4$

Try one yourself

Common questions

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

If it's written as one sandwich like $-2 \le x < 4$ , it's an AND — the value has to satisfy both ends at once. If the word 'or' appears between two separate inequalities, it's an OR. On a graph: one segment between two points is AND; two rays pointing away from each other is OR.

Can an AND inequality have no solution?

Yes. If the two pieces don't overlap — like $x < 1$ and $x > 5$ — no number satisfies both, so there's no solution. If you solve a sandwich and end with the larger number on the left, like $6 < x < 2$ , that's the signal there's no overlap.

Do I flip both signs when I divide a sandwich by a negative?

Yes — the flip rule applies to every inequality sign in the statement. After flipping, rewrite the sandwich so the smaller number sits on the left; $2 > x \ge -4$ and $-4 \le x < 2$ say exactly the same thing.

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Compound Inequalities

AND inequalities: the sandwich

OR inequalities: solve each piece separately

Reading and drawing the graphs

Worked examples

Example 1: a sandwich with addition

Example 2: a sandwich with two steps

Example 3: dividing a sandwich by a negative

Example 4: an OR inequality

Try one yourself

Common questions

Keep learning

Two-Step Inequalities

Absolute Value Equations

Equations with Variables on Both Sides