116 Chapter 2 Linear Equations and Inequalities in One Variable SECTION 2.6 Absolute Value Equations A memory card is measured with a ruler with centimeter gradations. The width is reported to be 2.5 cm. Because of the inaccuracy of measuring, the actual width is 2.5 ± 0.1 cm. This states that the distance between the exact width and the measured width is 0.1 cm. What is the actual width of the memory card? To solve this problem, we must solve the equation uw - 2.5u = 0.1, where w represents the exact width of the memory card. In this section, we will learn how to solve equations containing absolute values. Absolute Value Equations In Section 1.1, the absolute value of a number was presented. The absolute value of a number is the number’s distance from zero on a real number line. We can visualize this on a number line as shown. Note that both 5 and -5 are 5 units from zero. So, their absolute values are the same. 5 t f m 5 t f m –5 –4 –3 –2 –1 0 1 2 3 4 5 In fact, every real number and its opposite have the same absolute value, since they are the same distance from zero on a number line. Some examples are shown. u-5u = 5 and u5u = 5 u-6u = 6 and u6u = 6 u-7u = 7 and u7u = 7 u-4u = 4 and u4u = 4 u0u = 0 Important Facts About Absolute Value • The absolute value of any real number is nonnegative. • There are two numbers (a number and its opposite) that have the same absolute value. For instance, both -5 and 5 have the same absolute value, which is 5. • The number zero has an absolute value of zero. In this section, we will solve absolute value equations. Some examples are uxu = 5 uy - 4u = 7 u2x - 1u - 3 = 2 We will use the following property to solve absolute value equations. Property: Property 1 for Absolute Value Equations Let k be a real number. 1. If uXu = k and k > 0, then X = k or X = -k. 2. If uXu = 0, then X = 0. 3. If uXu = k and k < 0, then there are no solutions. Note: The expression inside the absolute value can be a single variable or a variable expression. Procedure: Solving an Absolute Value Equation Step 1: Isolate the absolute value on one side of the equation and the constant on the other side. Step 2: Set the expression inside the absolute value equal to the number(s) whose absolute value is the constant. ▶ OBJECTIVES As a result of completing this section, you will be able to 1. Solve absolute value equations of the form |X| = k, where k is a real number. 2. Solve absolute value equations of the form |X| = |Y|. 3. Solve applications of absolute value equations. 4. Troubleshoot common errors. Objective 1 ▶ Solve absolute value equations of the form |X| = k, where k is a real number.
hendricks_intermediate_algebra_1e_ch1_3
To see the actual publication please follow the link above