# Module 6: ## Module 6: Notes and Solutions to Select Practice Problems.

These solutions were prepared by Mr. John Hemmerling with revisions by Mr. Louis Aquila Practice Exercises-Unit 1. (3) -a - 3 = -10 Perform the opposite/inverse operation until you get the variable alone on one side of the equal sign. The opposite of -3 is +3 so: -a - 3 +3 = -10 +3 -a = -7 The variable is still not isolated. There is a minus sign next to it, so we are not finished. You can remove the minus sign by multiplying both sides of the equation by -1. (-1)(-a) =( -7)(-1) Thus, a = 7 Check: -a - 3 = -10 so -7 - 3 = -10 must be true.

√ (14)  3 e   9 5 There are two ways to solve this equation. 1) There are two operations here. The most 2) There is a fraction in front of the obvious is probably the division by 5. The opposite of division is multiplication so multiply everything by 5. 5 3 e  variable. Multiply a fraction by its reciprocal (Check out Module 5 in the Course Guide if you want to review this topic) and you get 1. Multiplying by 1 gives you the variable by itself. Since this -3e = -45 The only operation left is multiplication by (-3). The opposite of multiplication is is an equation, you must multiply everything by the reciprocal. 5   3   5 3 e  5  3 division so divide everything by (-3)  3 e  3   45  3 Thus e = 15 As you can see, the variable is alone now and all that's left is the simplification of the right-hand side (rhs) of the equation. e   9   3 5    45 3  15

Check:  3 e 5   9 so  3 ( 15 ) 5   45 5   9 (39) -14 = 5p + 18 This problem could just as easily been written as 5p + 18 = -14. Anyway, the solution of the original problem follows. -14 - 18 = 5p + 18 -18 Check: -14 = 5p + 18 so  14   14  5   32 5  32  5p 5 32  5 18  so p = -32/5 18 must be true. √ (47) 7  3 x 4  2 Observe that it makes no difference if the variable term is on the right or the left. 7  3

x

4  2 3

x

7 2 4 9  3

x

4 4    3

x

4 Check: 7  3 x  2 so 7  4 4 12 12 3

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 2 must be true. 7 7   36 9 4   2 2 √ (60) 2b - 9 = 5b This problem has the variable b on both sides of the equal sign. You have to get them together to solve the equation. Adding the opposite of either term to both sides of the equation will do it. 2b - 2b - 9 = 5b - 2b

 9  3 -9 = 3b 3 b 3 so b = -3 Check: 2b - 9 = 5b so 2(-3) - 9 = 5(-3) must be true. -6 -9 = -15 √ (68) y - (2 + 2y) = 6 First remove the grouping symbols. Remember to distribute the negative sign over every term within the parentheses. Then you need to combine like terms. y - (2 + 2y) = 6 y - 2 - 2y = 6 -2 - y = 6 -2 + 2 - y = 6 + 2 -y = 8 so y = -8 Check: y - (2 + 2y) = 6 so -8 - (2 + 2[-8]) = 6 must be true. -8 - (2 + [-16]) = 6 (74) -8 - (-14) = 6 -8 + 14 = 6 √  3 ( x  4 )   9 2 First, multiply everything by 2 to remove the denominator.  3(

x

2  4)    (2) We now have the equation: -3(x - 4) = -18 -3x + 12 = -18 -3x + 12 - 12 = -18 - 12 Check:  3 ( x 2  4 )  -3x = -30 x = 10  9 so  3 ( 10  4 ) 2   9  3 ( 10  must be true. 4 )   9  30 2  12   9 √  18 2  2  9