Compound Inequalities: And, and Or

In this lesson you will: Read and as the overlap of two conditions and or as everything in either one; Solve a three-part inequality by operating on all three parts at once; Graph both kinds of compound inequality on a number line.

And means both at the same time

Two conditions joined by and are satisfied only by numbers that meet both. On a number line that is the overlap of the two shaded regions, and for linear conditions the overlap is a single stretch.

x>-1 and x <= 4 together describe the numbers between -1 and 4, taking 4 but not -1. Written as a chain, that is -1<x <= 4.

Or means either one is enough

Two conditions joined by or are satisfied by any number that meets at least one of them. The graph is both regions, and for linear conditions they usually point away from each other with a gap between.

x<-2 or x >= 3 leaves everything from -2 to 3 unshaded. A number like 0 fails both conditions, so it is not a solution.

Three-part inequalities

An and statement can be written as a chain, and then you solve it by doing the same thing to all three parts at once. The middle is where the variable lives; the outside numbers come along for the ride.

For -5 <= 2x+1<9, subtract 1 from all three parts to get -6 <= 2x<8, then divide all three by 2 to get -3 <= x<4.

Common mistake — Writing an or statement as a chain

The answer is x<-2 or x>5, and it gets written as 5<x<-2.

The motive is tidiness. A chain looks like the compact, grown-up way to write a compound inequality, and the two numbers and two symbols are all present and correct.

But a chain asserts everything it contains, including the relationship between the two outer numbers. 5<x<-2 claims among other things that 5<-2. No number is both greater than 5 and less than -2, so the chain has an empty solution set — the exact opposite of the intended answer, which contains almost every number.

How to catch it: Read the two outer numbers of any chain you write, ignoring the middle. If the resulting statement is false — 5<-2 — the compound is an or and must stay as two separate inequalities joined by the word or.