Section 1.1 The Set of Real Numbers 9 ������������������������ The absolute value of a real number a is the distance between 0 and a on the real number line. The absolute value of a is denoted uau. –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 3 units 3 units From the graph, we see that the distance between 3 and 0 is 3 units and the distance between -3 and 0 is 3 units. We can state these facts as shown in the table. Verbal Statement Mathematical Statement The absolute value of three is three. u3u = 3 The absolute value of negative three is three. u-3u = 3 Because absolute value refers to distance, the absolute value of a number is always greater than or equal to zero. The following property summarizes this fact. �������������������� Absolute Value If a ≥ 0, then uau = a. If a < 0, then uau =-a. This property states that if a number is greater than or equal to zero, its absolute value is equal to that number. For instance, 5 ≥ 0 and 5 is 5 units from zero, so u5u = 5. If a number is less than zero, its absolute value is equal to the opposite of the number. For instance, -2 < 0 and -2 is 2 units from zero, so u-2u = 2. Note that 2 = -(- 2). ���������������������� Finding the Absolute Value of a Number Step 1: Find the number’s distance from zero. Step 2: The absolute value of a number is always greater than or equal to 0. �� ������������������������ ������������������ Simplify each absolute value expression. Problem Solution 5a. u6u u6u = 6, since 6 is 6 units from 0. 5b. u-8u u-8u = 8, since -8 is 8 units from 0. 5c. u0u u0u = 0, since 0 is 0 units from 0. 5d. P- 1 2 P `- 1 2 ` = 1 2 , since - 1 2 is 1 2 unit from 0. 5e. -u2u First, find the absolute value of 2, which is 2. Then find the opposite of 2, which is -2. -u2u =-(2)=-2 5f. -u-3u First, find the absolute value of -3, which is 3. Then find the opposite of 3, which is -3. -u-3u =-(3)=-3 Student Check 5 Simplify each absolute value expression. a. u10u b. u-6u c. uπu d. P- 2 5 P e. -u4u f. -u-7u
hendricks_beginning_algebra_1e_ch1_3
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