Inequalities joined by "and" or "or", how to solve them, how to graph the solution, and how to handle three-part inequalities.
Two conditions at once
"The temperature stayed above \(10\) degrees and below \(25\) degrees." "You get a discount if you're under \(12\) or over \(65\)." Each sentence combines two inequalities into one compound inequality. The word that joins them changes everything.
AND: where the graphs overlap
Three-part inequalities
An "and" inequality is often written in one line, like \(-3 \le 4x + 1 < 9\). Read it as "\(4x + 1\) is at least \(-3\)and less than \(9\)". You can solve it all at once by doing the same thing to all three parts.
OR: everything in either graph
Practice
Quick check
What is the solution of \(x > 1\) and \(x < 4\)?
"And" keeps only the overlap: numbers between \(1\) and \(4\), endpoints excluded.
Quick check
Which compound inequality describes "two rays pointing away from each other", from \(-2\) to the left and from \(2\) to the right, endpoints excluded?
Two separate pieces means a union, so the word is "or".
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