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4.3 Solving Compound Linear Inequalities
The words that we use can be very powerful and in fact, even the small
words, like “and” and “or” can change the entire meaning of a phrase
significantly when they are interchanged. Before we apply this to
mathematics, you can see it clearly in everyday language:
“Would you like to go to the beach or to the movies?”
“Would you like to go to the beach and to the movies?”
The only word that was changed in those two question was “or” to
“and”, but how would your response change? For the first question, you
might answer “the beach”. For the second question, you might answer
“yes”. They mean something different.
Here is another example:
“It is sunny and it is raining.”
“It is sunny or it is raining.”
Once again, the only word that has changed is the connecting word
“and” to “or”, but for the first statement to be true, both things have to
be true at the same time. It has to be sunny and it also has to be raining.
But for the second statement to be true, only one of them needs to be
true. If it is sunny, but not raining, the second statement is still true.
These statements mean something different.
These words are actually logical connectors and when applied to
statements, they form compound statements. In mathematics, when we
use them to join inequalities, we get compound inequalities. Now, let’s
apply this to mathematics. The thing to keep in mind is that “and”
means both have to be true and “or” means at least one of them is true.
For this reason, when we apply this to inequalities, the “and” means to
take the intersection or “overlap” since that is where both inequalities
will be true.
