184 CHAPTER 1 Equations and Inequalities Absolute Value Models for Distance and Tolerance If a and b represent two real numbers, then the absolute value of their difference, either a - b or b - a , represents the undirected distance between them. EXAMPLE 5 Using Absolute Value Inequalities with Distances Write each statement using an absolute value inequality. (a) k is no less than 5 units from 8. (b) n is within 0.001 unit of 6. SOLUTION (a) Since the distance from k to 8, written k - 8 or 8 - k , is no less than 5, the distance is greater than or equal to 5. This can be written as k - 8 Ú 5, or, equivalently, 8 - k Ú 5. Either form is acceptable. (b) This statement indicates that the distance between n and 6 is less than 0.001. n - 6 60.001, or, equivalently, 6 - n 60.001 S Now Try Exercises 69 and 71. EXAMPLE 6 Using Absolute Value to Model Tolerance In quality control situations, such as filling bottles on an assembly line, we often wish to keep the difference between two quantities within some predetermined amount, called the tolerance. Suppose y = 2x + 1 and we want y to be within 0.01 unit of 4. For what values of x will this be true? SOLUTION y - 4 60.01 Write an absolute value inequality. 2x + 1 - 4 60.01 Substitute 2x + 1 for y. 2x - 3 60.01 Combine like terms. -0.01 62x - 3 60.01 Case 2 2.99 6 2x 63.01 Add 3 to each part. 1.495 6 x 61.505 Divide each part by 2. Reversing these steps shows that keeping x in the interval 11.495, 1.5052 ensures that the difference between y and 4 is within 0.01 unit. S Now Try Exercise 75.
RkJQdWJsaXNoZXIy NjM5ODQ=