Section 1.1 Sets and the Real Numbers 11 ANSWERS TO STUDENT CHECKS Student Check 1 a. A = 54, 5, 6, 7, 8, . . .6 b. A B = 55, 10, 15, 20, 25, 30, 45, 60, 756 ; A B = 515, 306 c. i. A = 5UVA, UNC, W&M, GT, UCSB6 ii. B = 5U-M, W&M, UC Davis, UCSB6 iii. A B = 5W&M6, so UVA ∉ A B. Student Check 2 7.5 = 15 2 , rational, real; 3 4 5 = 19 5 , rational, real; 130 ≈ 5.48, irrational, real; 125 = 5 = 5 1 , natural, whole, integer, rational, real; -20=- 20 1 , integer, rational, real; 4π ≈ 12.57, irrational, real; 1 2 , rational, real Student Check 3 –5.4 √−3 25 –3 16 √⎯ –214 1 2 3 4 5 6 –6 –5 –4 –3 –2 –1 0 Student Check 4 a. -10 b. 14 c. 1 2 d. - 25 6 e. 16 f. -8.2 5. a. 12 b. 13 c. 3 5 d. -1 e. -1 SUMMARY OF KEY CONCEPTS 1. A set is a collection of objects. The union of two sets is a collection of all elements that are in either of the sets. The intersection of two sets is a collection of the elements that are in each of the sets. 2. The real numbers are comprised of rational numbers and irrational numbers. The set of rational numbers includes all integers. The set of integers includes all whole numbers. The set of whole numbers includes all natural numbers. 3. Every real number can be graphed on a real number line. Irrational numbers must be approximated to be graphed. 4. The numbers a and -a are opposites of one another. These numbers have the same distance from zero and are on opposite sides of zero. 5. The absolute value of a number measures the number’s distance from zero on the real number line. The absolute value of a number is either zero or positive. Vertical bars denote absolute value. GRAPHING CALCULATOR SKILLS We can use a calculator to approximate irrational numbers, find the opposite of a number, and find the absolute value of a number. Example 1: Approximate the value of 13 and π. Solution: Enter the square root of 3 using the second function and the squaring function. Enter π using second function and the carat symbol. 2nd x2 3 ) ENTER 2nd ENTER So, 13 ≈ 1.73 and π ≈ 3.14. Example 2: Find the opposite of -6 and 0.25. Solution: To find the opposite of a number, we use the symbol (–) not the subtraction sign. (–) ( (–) 6 ) ENTER (–) ( 2 5 ) ENTER So, -(-6) = 6 and -(0.25)=-0.25. Example 3: Find u-5u, u10u, and -u-7u. Solution: Enter the absolute value function by pressing the MATH menu and then choosing the NUM menu. MATH 1 (–) 5 ) ENTER MATH 1 1 ) 0 ENTER (–) MATH 1 (–) ) 7 ENTER So, u-5u = 5, u10u = 10, and -u-7u =-7.
hendricks_intermediate_algebra_1e_ch1_3
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