Asked by noah
An inequality using any of the inequality symbols has an infinite number of solutions. An inequality using
all but one of the inequality symbols also has an infinite number of nonsolutions. Which inequality
symbol produces an inequality with a finite number of nonsolutions?
all but one of the inequality symbols also has an infinite number of nonsolutions. Which inequality
symbol produces an inequality with a finite number of nonsolutions?
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Answered by
GPT-5 mini
Answer: the "≠" (not equal) symbol.
Reason: For each standard inequality symbol <, >, ≤, ≥, =, ≠ you can find an inequality with infinitely many solutions (e.g. x<1, x>−1, x≤1, x≥−1, x=x, x≠0). For every symbol except "≠" you can also make an inequality whose nonsolutions form an infinite set (e.g. for = use x=0, nonsolutions x≠0; for < use x<0, nonsolutions x≥0, etc.). But an inequality of the form x≠a fails only when x=a, a single (finite) value. Hence "≠" is the symbol that produces an inequality with a finite number of nonsolutions.
Reason: For each standard inequality symbol <, >, ≤, ≥, =, ≠ you can find an inequality with infinitely many solutions (e.g. x<1, x>−1, x≤1, x≥−1, x=x, x≠0). For every symbol except "≠" you can also make an inequality whose nonsolutions form an infinite set (e.g. for = use x=0, nonsolutions x≠0; for < use x<0, nonsolutions x≥0, etc.). But an inequality of the form x≠a fails only when x=a, a single (finite) value. Hence "≠" is the symbol that produces an inequality with a finite number of nonsolutions.
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