184 Chapter 2 Linear Equations and Inequalities TIP: Some textbooks use a closed circle or an open circle (● or ) rather than a bracket or parenthesis to denote inclusion or exclusion of a value on the real number line. For example, the solution sets for the inequalities are graphed here. 6 5 4 3 2 1 0 1 2 3 4 5 6 A statement that involves more than one inequality is called a compound inequality. One type of compound inequality is used to indicate that one number is between two others. For example, the inequality means that In words, this is easiest to understand if we read the variable first: x is greater than 2 and x is less than 5. The numbers satisfied by these two conditions are those between 2 and 5. 2 6 x and x 6 5. 2 6 x 6 5 73 c 73 6 5 4 3 2 1 0 1 2 3 4 5 6 x 7 1 x 7 1 and c 73 Graphing a Compound Inequality Example 2 ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ Graph the solution set of the inequality: Solution: means that 4.1 6 y 1.7 and 4 3 2 4.1 Shade the region of the number line greater than and less than or equal to 2. Set-Builder Notation and Interval Notation Graphing the solution set to an inequality is one way to define the set. Two other methods are to use set-builder notation or interval notation. Set-Builder Notation The solution to the inequality x 2 can be expressed in set-builder notation as follows: 5x 0 x 26 the set of all x such that x is greater than or equal to 2 1.7. 4.1 4.1 6 y y 1.7 4.1 6 y 1.7 ⎧⎪⎨⎪⎩ ⎧⎪⎨⎪ ⎧⎨⎩ ( 1 0 1 2 3 4 5 6 1.7 6 5 Skill Practice Graph the solution set. 4. 0 y 8.5 Classroom Example: p. 193, Exercise 12 Answer 4. 0 8.5
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