Section 1.7 Properties of Real Numbers 75 Troubleshooting Common Errors Some common errors associated with the properties of real numbers are shown next. �� ������������������������ ������������������ A problem and an incorrect solution are given. Provide the correct solution and an explanation of the error. a. Simplify (2x + 6) + 3. �������������������������������������� ���������������������������������������������������������������� To simplify this expression, we must apply the associative property of addition. The distributive property does not apply since 3 is connected by addition, not multiplication. 6 + 18 (2x + 6) + 3 2x + (6 + 3) 2x + 9 b. Simplify 4(6x + 7). �������������������������������������� ���������������������������������������������������������������� We must distribute 4 to each term in parentheses, not just the first term. 4(6x + 7) 24x + 28 Objective 4 ▶ Troubleshoot common errors. (2x + 6) + 3 6x 4(6x + 7) 24x + 7 ANSWERS TO STUDENT CHECKS Student Check 1 a. 10, - 1 10 b. - 7 8 , 8 7 c. -4y, 1 4y d. 9b, - 1 9b e. - a 3 , 3 a Student Check 2 a. x + 8 b. 15x c. b + 11 d. 18b Student Check 3 a. 5x + 30 b. 7x - 21 c. 6x + 2 d. -3x - 6 e. -6x + 24 f. 36x + 45y - 27 g. -8x - 2 h. -5x + 6 SUMMARY OF KEY CONCEPTS 1. The identity element is the number that leaves another number unchanged when an operation is performed on it. a. For instance, 0 is the identity element for addition since any number added to zero is the original number. b. Also, 1 is the identity element for multiplication since any number multiplied by 1 is the original number. 2. An inverse is a number which produces the identity element when an operation is performed on it. a. The additive inverse is the opposite of a number. b. The multiplicative inverse is the reciprocal of a number. 3. Addition and multiplication are both commutative and associative. We can change the order and the grouping of the numbers being added or multiplied and obtain the same result. 4. The distributive property enables us to multiply a group of numbers by another number. The outer number distributes to all terms inside parentheses. ) �� ���������������� �������� EXERCISE SET Write About It! Use complete sentences to explain the term or the process. 1. Identity element for addition 2. Identity element for multiplication 3. Additive inverse 4. Multiplicative inverse 5. Commutative property for addition 6. Commutative property for multiplication 7. Associative property for addition 8. Associative property for multiplication
hendricks_beginning_algebra_1e_ch1_3
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