Examples and Problems with Solutions
APPENDIX A: ANSWERS TO EXERCISES
Lesson 1
1 [pic], [pic], [pic], [pic][pic] are examples of algebraic expressions. [pic][pic] are examples of variables but also examples of algebraic expressions . Variables represent unknown numbers. If we know the value of [pic], we can evaluate[pic], and as a result we get a number.
2 a) “a squared” or “a raised to the second power.” b) “a cubed” or “a raised to the third power.” c) “a raised to the twelfth power.” d) ”2 to the m” or ” 2 raised to the m-th power.” e) “minus y” or “the opposite of y” f) “the product of c and d” or “c times d” g) “a minus b” h) “two-fifth times x” or “two-fifth x”
3 a) [pic] b) [pic] c) “there is no multiplication performed”
d) [pic] e) [pic] f) [pic]
4 a) multiplication b) division c) exponentiation d) division e) subtraction f) multiplication
5 a) Any time two operation signs are next to each other, parentheses are needed. b) If parentheses are
removed, only m would be raised to the fourth power. c) Any time two operation signs are next to each
other, parentheses are needed, even if the multiplication sign is not explicitly displayed d) If parentheses
are removed, only a would be raised to the fourth power. e) Any time two operation signs are next to
each other, parentheses are needed, even if the multiplication sign is not explicitly displayed f) Any time two operation signs are next to each other, parentheses are needed
6 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic] g) [pic] h) [pic] i) [pic]
7 a) [pic] b) [pic] c) multiplication
8 a) [pic]; b) [pic] c) [pic]
9 a) [pic] b) [pic] c)[pic] d) [pic]
10 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic] g) [pic] h) [pic] i) [pic]
11 a) [pic] b) [pic] c) [pic] d) [pic] f) [pic] g) [pic] h) [pic] i) [pic] j) [pic] k) [pic]
12 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic]
13 [pic]
14 [pic]
15 [pic]or [pic]
16 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic]
17 [pic] can not be evaluated with [pic], because the denominator of a fraction can not be 0.
If [pic], [pic] can be evaluated: [pic], but if [pic] then [pic] is not possible. Another example could be: [pic] cannot be evaluated with[pic]. (answers vary)
18 a) [pic] b) [pic] c) cannot be evaluated d) [pic] e) [pic] f) cannot be evaluated
19 a) [pic] b) [pic] c) [pic]
20 a) [pic] b) [pic]
21 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic]
22 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic]
23 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic]
24 a) [pic] b) [pic] c) [pic]
d) [pic] “not possible” e) [pic] f) [pic]
25 a) [pic] or [pic] b) [pic] c) [pic] d) [pic] e) [pic] or [pic] f) [pic] or [pic]
26 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic]
27 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic]
28 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic]
29 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic]
30 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic]
31 a) [pic] b) [pic] c) [pic] d) [pic] e) 20
32 a) [pic] b) [pic] c) [pic]
33 a) [pic] b) [pic] c) [pic]
34 a) [pic] b) [pic] c) [pic]
35 a)[pic] b) [pic] c) [pic]
36 a) [pic] b) [pic] c) [pic] d) [pic]
37 a) [pic] b) [pic] c) [pic]
38 [pic] if [pic], [pic] if [pic]; [pic] may be positive or negative depending on the value of [pic]
Lesson 2
1 a) [pic] b) [pic] c) [pic] d) [pic]
e) [pic] f) [pic]
2 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic]
f) [pic] g) [pic] h) [pic] i) [pic] j) [pic]
3 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic]
g) [pic] h) [pic] i) [pic] j) [pic] k) [pic] l) [pic]
4 [pic]
5 [pic]
6 [pic]
7 a) [pic] exponentiation b) [pic] addition
c) [pic] exponentiation d) [pic] opposite of x
e) [pic] subtraction f) [pic] division
g) [pic] multiplication h) [pic] division
8 a) Multiply x by 4 and then subtract y.
b) Add a and 3 and then divide the result by x.
c) Add x and 3 and then multiply the result by y.
d) Divide s by t and then add 2 to the result.
e) Square x and then multiply by 3.
f) Multiply x by 3 and then square the result.
g) Add a and c and then raise the result to the 4th power.
h) Raise c to the 4th power and then add the result to a.
9 a) parentheses are needed b) [pic] c) [pic] d) parentheses are needed
e ) parentheses are needed f) [pic] g) parentheses are needed h) [pic]
i) parentheses are needed j) parentheses are needed
10 a) [pic] b) [pic] c) [pic] d) Yes, because we performed the same operations. e) [pic]
11 a) [pic] b) [pic] c) [pic] cannot be performed
d) [pic] e) [pic] f) [pic] g) [pic]
12 a) [pic] b) [pic] [pic] d) [pic]
13 a) [pic] b) [pic] c) [pic]
d) [pic] e) [pic]
14 a) [pic] b) [pic] c) [pic] d) [pic]
15 a) [pic] b) [pic] c) [pic] d)[pic]
16 b) c) and f) We would have 0 in the denominator in these cases.
17 a) [pic] b) 60 c) [pic] d) [pic] e) [pic]
18 a) [pic] b) [pic] c) [pic]
d) [pic] e) [pic] cannot be performed .f) [pic]
19 a) [pic] b) [pic] c) [pic]
d) [pic] e) [pic]
20 a) [pic] b) 64 c) [pic] d) [pic] e) 1 f) [pic]
21 a) [pic] b) [pic]
22 a) [pic] b) 0.1 c) [pic]
23 a) [pic] b) [pic]
24 a) [pic] b) 0 c) 1.3 d) [pic] e) [pic][pic] f) [pic]
25 a) 4 b) 7 c) 0.5 d) 7 [pic] e) [pic] f) [pic]
26 a) 10 b) [pic] c) [pic]
27 a) [pic] b) [pic]
28 a) addition, [pic] b) multiplication c) multiplication , [pic] d) addition , [pic]
e) exponentiation, [pic] f) subtraction [pic]
29 a) [pic] b) [pic] c) [pic]
Lesson 3
1 In the expression [pic], [pic] and [pic] are called factors . In the expression [pic], [pic] and [pic] are called terms.
2 a) [pic] b) [pic] c) [pic] d) [pic]
3. All these expressions are equal (equivalent) because of the commutative property of addition.
4 a) [pic], [pic]; they are not equivalent. b) True
5 a) Terms: [pic]; [pic] b) Terms:[pic]; [pic]
c) Terms: [pic]; [pic] d) Terms: [pic] [pic]
e) Terms:[pic];[pic]
f) Terms: [pic];[pic]
6 a) [pic] b) [pic]
7 a) Terms: [pic][pic]; [pic] (answers vary)
b) Terms: [pic] ; [pic](answers vary)
8 (1) -- (C), (2) -- (E), (3) --(A) , (4) --(B), (5)-- (D
9 a) [pic]; factors: 2, a b) [pic]; factors: [pic]
c) [pic]; factors: [pic] d) [pic]; factors:[pic] 10 a) [pic] b) [pic] c) [pic] d) [pic]
11 a) [pic] b) [pic].
12 a) All these expressions are equal (equivalent) because of the commutative property of multiplication
b) [pic]; [pic] when [pic] and [pic], thus they are not equivalent,
13 They are all equivalent because of the commutative property of addition and multiplication
14 a) [pic] b) [pic] c) [pic]
d) [pic]
15 a) [pic] b)[pic] c) [pic] d) [pic]
16 a) [pic] b) [pic] c) [pic]
17 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f)[pic]
18 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic]
g) [pic] h) [pic]
19 The opposite to [pic] is [pic] or equivalently [pic], [pic]. All students were right.
20 a) [pic] b) [pic]
21 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic]
g) [pic] h) [pic]
22 a) [pic] b) [pic] c) [pic] d) [pic]
23 a) [pic] b) [pic] c) [pic] d) [pic] e) not possible f) not possible g) [pic] h) [pic] i) [pic] j) [pic] k) [pic] l) not possible m) [pic] n) [pic]
o) [pic] p)[pic] q) [pic] r) [pic] s) [pic] t) [pic]
24 Yes, both are equivalent.
25 [pic] and [pic] are not equivalent. In [pic]the entire expression of (x+y) is multiplied with ([pic]) and in [pic] only y is multiplied with ([pic]).
26 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic] g) [pic] h) [pic]
27 [pic], [pic] , [pic] are equivalent to [pic]
28 [pic] and [pic] are equivalent to [pic]
29 [pic]and [pic] are equivalent to [pic].
30 [pic] are equivalent to [pic].
31 All of the expressions are equal to [pic] except [pic]
32 Determine which of the following expressions are equivalent to [pic] :
[pic], [pic], and [pic] are equivalent to [pic]
33 [pic], [pic], [pic] are equivalent to [pic]
34 [pic]are equivalent to [pic].
35 [pic] are equivalent to [pic].
36 Both John and Mary are right.
37 [pic], [pic]. Since [pic] the expressions are not equivalent.
38 [pic], [pic], when [pic], [pic]. Since [pic] the expressions are not equivalent.
39 [pic], [pic], when [pic],[pic],and [pic]. Since[pic] the expressions are not equivalent.
40 a) [pic] [pic], when [pic] b) [pic] [pic], when [pic]
c) [pic] [pic], when [pic] d) [pic] [pic], when [pic]
e) No, we cannot. Even if the expressions have the same answers in a-d, we cannot conclude that the expression will always be equivalent. f) [pic] [pic], when [pic]. Yes, we can, they are not equivalent. It is enough to find one set of values of variables for which two expressions are not equal to determine that they are not equivalent.
Lesson 4
1 .coefficient, exponent or power, the base.
2 a) first b) zero
3 a) b b) ab c) de d) ─a e) a f) [pic]
4 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic] g) [pic]
5 base exponent coefficient
a) [pic] 4 3
b) [pic] m [pic]
c) [pic] 3 [pic]
d) [pic] 2 [pic]
e) [pic] m 1
f) [pic] 7 [pic]
g) [pic] 7 [pic]
h) [pic] 5 [pic]
6 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic]
g) [pic] h) [pic] i) not possible j) [pic] k) [pic]
l) [pic] m) [pic] n) [pic] o) [pic] p) [pic]
q) [pic] r) [pic] s) [pic] t) [pic]
7 a) [pic] b) [pic] c) [pic] d) [pic]
e) [pic] f) [pic] g) [pic] h) [pic]
8 a) 1 b) 3 c) [pic] d) [pic] e) [pic] f) 1 g) [pic] h) 1
9 a) [pic] b) [pic], necessary c) [pic] d) [pic], necessary
e) [pic], necessary f) [pic] g) [pic], necessary h) [pic], necessary
10 a) [pic] b) [pic]
The answers are different, since the order of operations is different. In part b, we must first complete operations within parentheses.
11 a) To multiply exponential expressions with the same bases one needs to add the exponents.
b) To divide exponential expressions with the same bases one needs to subtract their exponents
c) To raise an exponential expression to another power one needs to multiply exponents.
12 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic] g) [pic] h) 1
i) [pic] j) [pic] k) [pic] l) [pic] m) [pic] n) [pic] o) [pic] p) [pic]
13 a) [pic] b) [pic] c) [pic]
14 a) [pic] b) [pic] c) [pic] d) [pic]
15 a) [pic] b) [pic] c) [pic] d) [pic]
16 a) [pic] b) [pic] c) [pic] d) [pic]
17 a) [pic] nc: [pic] b) [pic] nc: [pic] c) [pic] nc: [pic] d) [pic] nc: [pic]
18 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic] g) [pic] h) [pic] i) [pic] j) [pic] k) [pic] l) 1 m) [pic] n) [pic] o) [pic] p) [pic]
19 a) [pic] b) [pic] c) [pic]
20 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic] g) [pic] h) [pic] i) [pic] j) [pic] k) [pic] l) [pic]
21 a)[pic] b) [pic]
22 a) [pic] b) [pic] c) [pic] d) 1
23 [pic], [pic], [pic], [pic], [pic]
24 [pic], [pic], [pic], [pic], [pic]
25 [pic], [pic], [pic] [pic], [pic]
26 [pic], [pic], [pic], [pic], [pic][pic]
27 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic]
28 a) 15 b) [pic] c) [pic] d) [pic] e) [pic] f) [pic]
Lesson 5
1 Based on the Commutative Law of Multiplication: [pic], and from here we can apply the Distributive Law: [pic].
2 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic]
g) [pic] h) [pic]
3 Based on the Commutative Law of Addition[pic], so the two answers are equivalent. Some other ways (answers vary):
[pic]4 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic] g) [pic] h)[pic] i) [pic] j)[pic]
k) [pic] l) [pic]
5 a) [pic] b) [pic] c) [pic]
d) [pic] e) [pic]
f) [pic]
6 a) [pic] (answers vary)
b) [pic] (answers vary)
7 .[pic] [pic] [pic] [pic] [pic]
8. [pic] [pic] [pic] [pic]
[pic] [pic] [pic] [pic]
9 a) 2 b) 3 c) m
10 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic] g) [pic]
11 a) [pic] b) [pic] c) [pic]
12 a) [pic] b) [pic] c) [pic]
13 a) [pic] b) [pic] c) [pic] d) [pic]
e) [pic] f) [pic] g) [pic] h) [pic]
14 a) [pic] b) [pic] c) [pic]
15 a) [pic] b) [pic]
16 a) [pic] b) [pic] c) [pic]
d) [pic] e) [pic] f) [pic]
g) [pic] h) [pic]
17 [pic]
18 a) [pic] b) [pic] c) [pic] d) [pic]
19 a) Numerator: One term, t. (can be viewed as two factors, 1 and t) Denominator: two terms: 2t (with factors 2 and t) and ─ty (with factors ─1, t and y). Therefore t is a common factor of ALL terms (numerator and denominator.) We can divide the numerator and the denominator by t.
b) Numerator: two terms, x (with factors 1 and x) and xy (with factors x and y) Denominator: one term, 2ax (with factors 2, a and x). ALL terms in the numerator AND in the denominator have a common factor, x. We can therefore divide both the numerator and denominator by x.
c) Numerator: one term, 3ab (with three factors 3, a and b). Denominator: two terms, ab (with factors a and b) and ─[pic] (with factors ─1 and [pic]). ALL terms in the numerator AND in the denominator have a common factor, a. We can therefore divide both the numerator AND the denominator by a.
20 x is NOT a factor in the denominator, but it can be viewed as a factor in the numerator. We can NOT cancel x, because it is not a factor in the denominator.
21 x IS a factor in BOTH the numerator and the denominator, therefore we can cancel x.. The result is [pic]
22 a IS a factor in BOTH the numerator and the denominator, therefore we can cancel it. The result: [pic]We can not cancel it.
23 [pic] IS a factor in the denominator, but NOT in the denominator, therefore we cannot cancel it.
24 a) [pic] ; 3xy b) [pic] ; 2ab c) [pic] ; [pic] d) [pic]; [pic] e) [pic] ; [pic] f) [pic]; a g) [pic]x h) [pic]; b i) [pic] ; [pic] j) not possible
25 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic] g) [pic] h) [pic] i) not possible j) [pic] k) not possible l)[pic] m) [pic] n) [pic] o) [pic] p) [pic]
Lesson 6
1 b) and d)
2 Yes.
3 No. (All three are unlike)
4 [pic] [pic] [pic] [pic]
5 [pic] [pic] [pic] [pic]
6 [pic] [pic] [pic] [pic] [pic]
7 [pic] [pic] [pic] [pic] [pic]
8. a) [pic] b) not possible c) 0 d) not possible e) [pic] f) [pic] g) not possible h) [pic] i) [pic] j) [pic] k) not possible l) [pic]
9 a) [pic] b) [pic] c) [pic]
d) [pic] e) [pic] f) [pic]
10 a) [pic] b) [pic] c) x d) [pic] e) [pic] f) not possible g) not possible h) [pic]
11 a) [pic] b) [pic]
c) [pic]
12 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic]
g) [pic] h) [pic] i) [pic] j) [pic] k) [pic]
l) [pic]
13 Student A and C.
14 [pic]; a) [pic] b) [pic] c) [pic]
15 [pic]; a) [pic] b) [pic] c) [pic].
16 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic] g) [pic] h) [pic] i) [pic] j) [pic] k) [pic] l) [pic] m) [pic]
n) [pic] o) [pic] p) [pic] q) [pic] r) [pic]
17 a) The student claiming that [pic] was right.
b) [pic]
18 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic] g) [pic] h) [pic] i) [pic] j)[pic]
19 a) [pic] b) [pic]
c) [pic] d) [pic]
e) [pic] f) [pic]
g) [pic] h) [pic]
i) [pic] j) [pic]
20 [pic] a) 9 b) [pic] c) 5.4
21 [pic] a) [pic] b) [pic] c) [pic]
Lesson 7
1 a) 20 b) [pic] c) [pic]
2 a) 6 b) [pic] c) 4 d) [pic]
3 a) [pic] b) [pic] c) 4
4 a) [pic] b) [pic] c) 2 d) 4
5 a) [pic] b) [pic] c) [pic] d) [pic]
6 a) [pic] b) 7 c) [pic] d) [pic] e) [pic]
7 a) 1 b) [pic] c) [pic] d) [pic]
8 a) 9 b) [pic] c) 3 d) [pic] e) 6 f) 27
9 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic]
10 a) 1 b) 2 c) [pic]
11 a) [pic] b) [pic] c) [pic]
12 a) [pic] b) [pic] c) [pic] d) [pic]
13 a) [pic] b) [pic] c) [pic] d) [pic]
14 a) [pic] b) [pic] c) 0.05 or [pic] d) 10
15 a) [pic] b) [pic]
16 a) [pic]
17 a) [pic] b) [pic] c) [pic] d) [pic]
18 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic]
19 a) [pic] b) [pic] c) [pic] d) 0
20 a) [pic] b) [pic] c) [pic]
21 a) [pic] b) [pic] c) [pic] d) [pic]
22 [pic]
23 a) [pic] b) [pic]
24 a) [pic] b) [pic] c) [pic]
25 a) [pic] b) [pic] c) [pic]
26 a) [pic] b) [pic] c) equal d) 0 e) [pic] f) equal
27 [pic]
28 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic]
29 a) [pic] b) [pic] c) y d) [pic]
30 [pic]
Lesson 8
1 One can solve an equation but not an algebraic expression_. If the left hand side of an equation is equal to the right hand side of the equation for [pic], then 7 is called a _solution_. The solutions_ of an equation are all values of variables that make the equation true. The statement that contains two quantities separated by an equal sign is called an _equation . A _solution__ always makes the _equation_ true.
2. Equations: b) [pic] c) [pic] e) [pic]
3 Both Tom’s and Mary’s answers are correct, because [pic] is equivalent to both
[pic] and [pic].
4 False. 7 is not a solution of [pic].
5 None of the numbers is a solution of [pic]. The number 2 and [pic] are solutions of [pic].
6. a) No b) Yes
7 a) Yes b) No c) Yes
8 a) No b) Yes c) Yes d) Yes
9 a) No b) No c) Yes
10 a) No b) Yes c) Yes d) No
11 a) No b) Yes c) Yes d) Yes
12 a) Yes b) Yes
13 a) Yes b) Yes
14 a) [pic] b) 5 (answers vary.)
15 For example [pic] (answers vary)
16 For example [pic] is a solution, and [pic] is not a solution. (answers vary)
17 We can only divide by a variable if we assume it is not 0. It is better if we always try to avoid dividing by a variable or by an algebraic expression. A better way to solve this equation is: [pic]
[pic] [pic]
18 a) [pic] b) [pic] c) [pic] d) [pic]
e) [pic] f) [pic] g) [pic] h) [pic]
19 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic]
20 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic]
21 f) [pic] g) [pic] c) [pic] h) [pic] e) [pic] f) [pic]
22 a) no solution b) [pic] (exactly one solution) c) all real numbers
d) no solution e) all real numbers
23 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f)[pic]
g) [pic] h) [pic] i) [pic] j) [pic] k) [pic]
l) [pic] m) [pic] n) [pic] o) all real numbers p) [pic]
q) [pic] r) no solution s) no solution t) no solution u) [pic] v) [pic]
Lesson 9
1 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic]
2 a) no solution b) [pic] c) [pic] d) [pic] e) no solution
f) [pic] g) [pic] h) [pic] i) all real numbers
j) [pic] k) [pic] l) [pic]
3 a) [pic] b) [pic] c) [pic]
4 a) [pic] b) [pic] c) [pic]
5 a) no solution b) [pic] c) [pic]
6 No. [pic] is not an equation, so we can not solve it.
7 a) [pic] b) [pic]
8 a) [pic] b) [pic]
9 a) [pic] b) [pic]
10 a) [pic] b) [pic]
11 a) [pic] b) [pic]
12 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic] g) [pic] h) [pic] i) [pic] j) [pic] k) [pic] l) [pic] m) [pic] or [pic] n) [pic] o) [pic]
13 a) [pic] b) [pic] c) [pic]
14 a) [pic] b) [pic] or [pic] c) [pic] or [pic]
15 [pic]
16 [pic]
17 a) [pic] b) [pic]
18 a) [pic] or [pic] b) [pic] c) [pic] inches
Lesson 10
1 For example [pic] (answers vary). The same numbers could be solutions for the other inequality. Both xx state the same condition (are equivalent)
2 a) and d)
3 For example [pic] (answers vary.)
4 [pic]
5. [pic], [pic], [pic], 0, 1, 2, 3, 4, 5
6. [pic], [pic], [pic], 0, 1, 2, 3, 4, 5
7 [pic]
8 a) [pic] b) [pic] c) [pic] d) [pic] e) [pic]
9 a) [pic]
[pic]
b) [pic]
[pic]
c) [pic]
[pic]
10 a) All numbers that are at least [pic]
[pic]
b) All numbers no more than 4
[pic]
c) all non-negative numbers
[pic]
d) All numbers that are at most [pic]
[pic]11 a) [pic] b) [pic] c) [pic] d) [pic]
12 [pic] is the value that satisfies both inequalities.
[pic]
13 a) [pic] (answers vary)
[pic] b) [pic] (answers vary)
[pic]
14 a) [pic] (answers vary)
[pic]
b) [pic] (answers vary)
[pic]
15 [pic] (answers vary.)
16 [pic] (answers vary)
17 a) [pic] and b) [pic] ; [pic] is not a solution of (c) and (d)
[pic] [pic]
18 Only student B
19 a) yes b) no (there are infinitely many solutions) c) infinitely many
d) 1, 10, 100 e) [pic] (answers vary.) f) yes g) no h) no i) yes
20 a) and e)
21 a) subtract 5; no sign change; [pic] b) add 2; no sign change; [pic]
c) divide by 4; no sign change; [pic] d) divide or multiply by -1; sign changes; [pic]
e) multiply by -3; sign changes; [pic]
22 a) and d)
23 b) and d)
24 a) [pic]
[pic]
b) All real numbers
[pic]
c) [pic]
[pic]
d) [pic]
[pic]
e) [pic]
[pic]
f) [pic]
[pic]
25 a) “add 3 to both sides”, “divide each side by [pic]”; [pic]
b) “subtract 1 from both sides”, “divide each side by 3”; [pic]
c) “multiply each side by 4”, “divide each side by 3”; [pic]
d) “multiply each side by 4”, “divide each side by [pic]”; [pic]
e) “subtract 1 from both sides”, “multiply each side by 4”; [pic]
f) “multiply each side by 4”, “subtract 1 from both sides”; [pic]
26 a) [pic] b) [pic] c) [pic] d) all real numbers e) no solution
f) [pic] g) [pic] h) [pic] i) [pic] j) all real numbers
k) all real numbers l) [pic] m) all real numbers n) no solution o) no solution p) [pic] q) [pic] r) [pic] s) [pic] t) no solution
Lesson 11
1 [pic]
2 a) [pic] b) [pic]
3 a) [pic] b) [pic]
4 [pic]
5 [pic]
6 [pic]
7 a) [pic] b) [pic]
8 a) [pic] b) [pic]
9 (1)-E; [pic] (2)-F; [pic] (3)-D; [pic]
(4)-A; [pic] (5)-B; [pic] (6)-C; [pic]
10 a) form of[pic]; [pic] b) form of [pic]; [pic]
c) form of [pic]; [pic] d) form of [pic]; [pic]
11 a) [pic] b) [pic] c) [pic]
d) [pic] e) [pic] f) [pic] [pic]
12 a) [pic] b) [pic] c) [pic] d) [pic]
e) [pic] f) [pic] g) [pic] h) [pic]
i) [pic] j) [pic]
13 a) [pic] b) [pic] c) [pic] d) [pic]
e) [pic] f) [pic] g) [pic] h) [pic]
i) [pic] j) [pic]
14 a) [pic] b) [pic] c) [pic]
15 a) [pic] b) [pic] c) [pic] d) [pic]
16 a) [pic] b) [pic] c) [pic]
d) [pic] e) [pic] f) [pic]
17 a) [pic] b) [pic]
c) [pic] d) [pic]
e) [pic] or[pic] if you were to multiply both sides by 4, then [pic]
f) [pic] or [pic]
18 a) [pic] b)[pic] c) [pic]
d) [pic] e) [pic] f) [pic]
19 a) [pic] b) [pic]
c) [pic] d) [pic]
e) [pic] f) [pic]
g) [pic] h) [pic]
20 a) [pic]
b) [pic]
c) [pic]
21 a) not linear b) [pic] c) [pic]
d) [pic] or [pic] e) [pic] , [pic]
f) [pic] or [pic]
g) [pic]
h) [pic] or[pic]
22 [pic] ; [pic] where [pic]
23 a) [pic] b) [pic] c) [pic] d) [pic]
e) [pic] f) [pic] g) [pic]
h) [pic]
i) [pic]
j) [pic]
24 a) [pic]
b) [pic]
c) [pic]
d) [pic]
25 a) [pic] b) [pic]
c) [pic] d) [pic]
26 [pic]
27 [pic]
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