Maintaining Mathematical Proficiency
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There are two types of compound inequalities. How do they manifest on a numberline?
Yes.
Explanation: See solution.
There are two types of compound inequalities.
Andcompound inequalities
Orcompound inequalities
If an inequality has all real numbers as a solution, it has to cover every number on the numberline.
Let's analyze the different types of compound inequalities.
Andcompound
The first type describes an interval between two values. For example, the compound inequality less than 2 and greater than - 3 is mathematically written as - 3
We see that it covers the area between - 3 and 2 on the numberline. Therefore, this type of an inequality will never cover the entire numberline as you can always pick a number that's outside of its two endpoints.
Orcompound
The second type describes two separate intervals that run in opposite directions. For example, the compound inequality no more than - 2 or no less than 2 is mathematically written as x≤ - 2 or x≥ 2.
In this example, you could pick a value on the numberline that's between the inequalities.
However, each inequality run infinitely in the opposite direction. Therefore, to cover the entire numberline, all we need to do is to reduce the distance between them to 0. For example, the inequality less than 0 or greater than or equal to 0 is an example of an inequality that covers the entire numberline.
