6.5 Solving Linear Inequalities 331 An inequality consists of two (or more) expressions joined by an inequality symbol. Examples of inequalities < < − ≥ x x 3 5, 2, 3 2 5 A statement of inequality can be used to indicate a set of real numbers. For example, x 2 < represents the set of all real numbers less than 2. Listing all these numbers is impossible, but some are 2, 1.234, 1, , 0, , 1. 1 2 97 163 − − − − Solving Linear Inequalities To indicate all real numbers less than 2, we can use the number line. The number line was discussed in Chapter 5. To indicate the solution set of x 2 < on the number line, we draw an open circle at 2 and a line to the left of 2 with an arrow at its end. This technique indicates that all points to the left of 2 are part of the solution set. The open circle indicates that the solution set does not include the number 2. 25 24 23 22 21 0 1 2 3 4 5 x < 2 To indicate the solution set of x 2 ≤ on the number line, we draw a closed (or darkened) circle at 2 and a line to the left of 2 with an arrow at its end. The closed circle indicates that the 2 is part of the solution. 25 24 23 22 21 0 1 2 3 4 5 x 2 Example 1 Graphing an Inequality Graph the solution set of ≤ x 1 on the number line. Solution The numbers less than or equal to 1 are all the points on the number line to the left of 1 and 1 itself. The closed circle at 1 shows that 1 is included in the solution set. 25 24 23 22 21 0 1 2 3 4 5 x 1 7 Now try Exercise 9 The inequality statements x 2 < and x 2 > have the same meaning. Note that the inequality symbol points to the x in both cases. Thus, one inequality may be written in place of the other. Likewise, x 2 > and x 2 < have the same meaning. Note that the inequality symbol points to the 2 in both cases. We make use of this fact in Example 2. Example 2 Graphing an Inequality Graph the solution set of < x 3 on the number line. Solution We can restate x 3 < as x 3. > Both statements have identical solutions. Any number that is greater than 3 satisfies the inequality x 3. > The graph includes all the points to the right of 3 on the number line. To indicate that 3 is not part of the solution set, we place an open circle at 3. 23 22 21 0 1 2 3 4 5 6 3 < x or x > 3 7 Now try Exercise 11
RkJQdWJsaXNoZXIy NjM5ODQ=