Using trichotomy to solve inequalities
The three parts of the idea of “trichotomy”
How can you describe the relationship between two numbers? There are only three ways to describe this relationship, which is why it’s called the law of trichotomy (“tri” means three).
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The law of trichotomy says that two numbers can have exactly one of three possible relationships:
1. The first value is smaller than the second value, ???a<b???.
2. The first value is greater than the second value, ???a>b???.
3. The first value is equal to the second value, ???a=b???.
Because of the trichotomy law, we can make the following three statements:
If ???a??? is not greater than ???b??? and also not equal to ???b???, then ???a??? must be less than ???b???. If ???a\ngeq b???, then ???a<b???.
If ???a??? is not less than ???b??? and also not equal to ???b???, then ???a??? must be greater than ???b???. If ???a\nleq b???, then ???a>b???.
If ???a??? is not greater than ???b??? and also not less than ???b???, then ???a??? must be equal to ???b???. If ???a\nless b??? and ???a\not>b???, then ???a=b???.
How to use the idea of trichotomy to define relationships between numbers
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Solving inequalities using the idea of trichotomy
Example
Solve the inequality.
???4(x-2)\ngeq 3(x+8)???
We need to recognize that, due to the trichotomy law, if a value is not greater than or equal to another value it must be less than the other value. Therefore we can rewrite ???4(x-2)\ngeq 3(x+8)??? with a less than sign.
???4(x-2)<3(x+8)???
We basically follow the same steps to solve inequalities as we do equations, but don’t accidentally change your ???<??? to ???=???. We’ll simplify both sides of the inequality by distributing.
???4x-8<3x+24???
Now move the ???3x??? so that all the ???x??? terms are on the left side.
???4x-3x-8<3x-3x+24???
???x-8<24???
Add ???8??? to both sides.
???x-8+8<24+8???
???x<32???
Let’s try another example of trichotomy.
The law of trichotomy says that two numbers can have exactly one of three possible relationships.
Example
Solve the inequality.
???4x+5\nleq2x+7???
This time, let’s solve for ???x??? first, just so that we can see that we can do these kinds of problems in any order that we like. Start by subtracting ???2x??? from both sides so that all ???x??? terms are on the left side.
???4x-2x+5\nleq2x-2x+7???
???2x+5\nleq7???
Subtract ???5??? from both sides.
???2x+5-5\nleq7-5???
???2x\nleq2???
Divide both sides by ???2???.
???\frac{2x}{2}\nleq\frac{2}{2}???
???x\nleq1???
Since ???x??? isn’t less than ???1???, and isn’t equal to ???1???, it can only be greater than ???1???, according to the trichotomy law.
???x>1???
