To decide how to determine whether an absolute valueinequality uses a compound inequality with and or a compound inequality with or, let's consider different absolute value inequalities. Let n be any number greater than or equal to 0. If we take the absolute value expression |x|, we can write two simple inequalities.
First Inequality:&|x|< n
Second Inequality:&|x|>n
Let's solve them!
First Inequality
To solve this inequality, we have to consider two cases, a non-negative and negative one.
Case 1:&x< n
Case 2:&x> - n
We can tell that the first inequality is true when x is less than n, and the second one is true when x is greater than - n. Therefore, we can graph the solution set on the number line as shown below.
As we can see, the solution set is all real numbers greater than - n and less than n. Therefore, it is a compound inequality with and. Notice that if the inequality symbol was not strict, it would still be the same kind of inequality.
Second Inequality
To solve this inequality, we also have to consider two cases, a non-negative and negative one.
Case 1:&x> n
Case 2:&x< - n
We can tell that the first inequality is true when x is greater than n, and the second one is true when x is less than - n. Therefore, we can graph the solution set on the number line as shown below.
As we can see, the solution set is all real numbers less than - n or greater than n. Therefore, it is a compound inequality with or. Notice that if the inequality symbol was not strict, it would still be the same kind of inequality.
Conclusion
We can conclude that when an absolute value is on the left and the inequality symbol is < or ≤, the compound sentence uses and, and if the inequality symbol is > or ≥, the compound sentence used or.
