Compound Inequalities

Compound Inequalities

A compound inequality combines two inequalities into a single statement using the word "and" or the word "or". The word used changes what the solution set looks like and how you should interpret it.

"And" Compound Inequalities

An "and" inequality requires both conditions to be true at once. It is often written as a single chained expression, like -2 < x \leq 5, which means "x is greater than -2 and x is less than or equal to 5." The solution set is the overlap (intersection) of the two individual solution sets — a bounded segment of the number line.

"Or" Compound Inequalities

An "or" inequality requires at least one of the two conditions to be true. For example, x < -1 \text{ or } x \geq 4 describes two separate rays on the number line that don't connect — any number satisfying either piece is a solution. The solution set is the union of the two individual solution sets.

Solving a Chained "And" Inequality

To solve a chained inequality like -2 < 3x + 1 \leq 10, perform the same operation on all three parts at once (left, middle, and right).

Example

Solve -2 < 3x + 1 \leq 10.

1. Subtract 1 from all three parts: -2 - 1 < 3x \leq 10 - 1 \;\Rightarrow\; -3 < 3x \leq 9
2. Divide all three parts by 3 (positive, so no sign flip): -1 < x \leq 3

The solution is every x strictly greater than -1 and less than or equal to 3.

Interval Notation

Interval notation is a compact way to write a solution set as a pair of endpoints.

• Use a parenthesis ( or ) next to an endpoint that is not included (strict inequality).
• Use a bracket [ or ] next to an endpoint that is included.
• The solution -1 < x \leq 3 is written (-1, 3].
• An "or" solution like x < -1 \text{ or } x \geq 4 is written as a union: (-\infty, -1) \cup [4, \infty).

Key Takeaways

• "And" inequalities need both conditions true — the solution is a single connected interval.
• "Or" inequalities need at least one condition true — the solution can be two separate pieces.
• Solve a chained "and" inequality by applying the same operation to all three parts.
• In interval notation, use brackets for included endpoints and parentheses for excluded ones.