This chapter takes the previous chapter one step further -- to compound inequalities, or inequalities made up of two or more separate inequalities.
The notion of compound inequalities is complex. In order to understand how two inequalities interact, we must first understand how sets of numbers interact in general. Thus, the first section is an introduction to sets, starting with an explanation of set notation. It goes on to explain how two sets interact, which is demonstrated visually in a Venn diagram.
By using Venn diagrams, we learn how to find the union and intersection of two finite sets of numbers. These are discussed in the second section.
After learning how sets interact, we can better understand how solution sets of inequalities interact. This is the topic of the third section. This section demonstrates how to graph the union of inequalities and the intersection of inequalities.
The final section brings the material learned in the first three sections together, teaching how to solve and graph compound inequalities. They are graphed either as the union of two inequalities or the intersection of two inequalities.
The main goal of this chapter is to explain compound inequalities, which appear frequently in algebra and geometry. In addition, this chapter teaches important facts about sets of numbers. Mathematicians use sets both to classify numbers and to discover interesting facts about them; in fact, set theory is an entire branch of mathematics. The elementary knowledge of sets gained from this chapter will be of great use in all areas of math.
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