Compound Inequalities

AND Compound Inequalities

An AND inequality requires both conditions to be true simultaneously. The solution is the intersection (overlap) of the two solution sets.

Double inequality form: $a < x < b$ means "$x > a$ AND $x < b$."

Example: $-3 \leq 2x + 1 < 7$

Solve all three parts simultaneously:

Subtract 1: $-4 \leq 2x < 6$

Divide by 2: $-2 \leq x < 3$

Solution: $-2 \leq x < 3$

OR Compound Inequalities

An OR inequality requires at least one condition to be true. The solution is the union of the two solution sets.

Example: $x < -1$ OR $x > 4$

The solution includes all numbers less than $-1$ and all numbers greater than 4.

Solving Double Inequalities

Perform the same operation on all three parts:

$5 \leq 3x - 1 \leq 14$

Add 1: $6 \leq 3x \leq 15$

Divide by 3: $2 \leq x \leq 5$

Key rule: If you multiply or divide by a negative, flip BOTH inequality signs.

$-8 \leq -2x \leq 4$

Divide by $-2$ (flip both): $-2 \leq x \leq 4$ → Wait, let me redo:

$-8 \leq -2x$ becomes $4 \geq x$ and $-2x \leq 4$ becomes $x \geq -2$

So: $-2 \leq x \leq 4$

Finding the Range of an Expression

If $1 < x < 5$, what is the range of $3x + 2$?

Multiply by 3: $3 < 3x < 15$

Add 2: $5 < 3x + 2 < 17$

Tip 1: Treat the Double Inequality as One Step

Whatever operation you perform, apply it to all three parts simultaneously.

Tip 2: Be Extra Careful with Negatives

When multiplying/dividing by a negative in a double inequality, flip both inequality signs and reverse the order of the bounds.

Tip 3: Separate If Needed

If a compound inequality is confusing, split it into two separate inequalities, solve each, then combine.

Tip 4: Watch for Empty Sets

$x > 5$ AND $x < 3$ has no solution — no number is simultaneously greater than 5 and less than 3.

Tip 5: Context Problems Often Use AND

"Between 10 and 50" means $10 \leq x \leq 50$ (an AND compound inequality).