 1.2.1: 16 + (30) ______________
 1.2.2: 16 + 16 ______________
 1.2.3: (16) ______________
 1.2.4: 16 ______________
 1.2.5: 16  (30) ______________
 1.2.6: (16) 3 ______________
 1.2.7: (16)(30) ______________
 1.2.8: To simplify 4  49 + 8 , 4 # 2, first ______________.
 1.2.9: To simplify 15  [3  (1)] + 12 , 2 # 3, first ______________.
 1.2.10: a , called the ______________ of a, represents the distance from __...
 1.2.11: If a 0, then a = ______________. If a 6 0, then a = ______________.
 1.2.12: a + (a) = ______________: The sum of a real number and its additiv...
 1.2.13: If a and b are real numbers, the commutative property of addition s...
 1.2.14: If a, b, and c are real numbers, the associative property of multip...
 1.2.15: If a, b, and c are real numbers, the distributive property states t...
 1.2.16: An algebraic expression is ______________ when parentheses have bee...
 1.2.17: 6.8 + 2.3
 1.2.18: 7.9 + 2.4
 1.2.19: 11 15 + a 3 5 b
 1.2.20: 7 10 + a 4 5 b
 1.2.21:  2 9  3 4
 1.2.22: 3 5  4 7
 1.2.23: 3.7 + (4.5)
 1.2.24: 6.2 + (5.9)
 1.2.25: 0 + (12.4)
 1.2.26: 0 + (15.3)
 1.2.27: 12.4 + (12.4)
 1.2.28: 15.3 + (15.3)
 1.2.29: x = 11
 1.2.30: x = 13
 1.2.31: x = 5
 1.2.32: x = 9
 1.2.33: x = 0
 1.2.34: x =  22
 1.2.35: 3  15
 1.2.36: 4  20
 1.2.37: 8  (10)
 1.2.38: 7  (13)
 1.2.39: 20  (5)
 1.2.40: 30  (10)
 1.2.41: 1 4  1 2
 1.2.42: 1 10  2 5
 1.2.43: 2.3  (7.8)
 1.2.44: 4.3  (8.7)
 1.2.45: 0  ( 22)
 1.2.46: 0  ( 23)
 1.2.47: 9(10)
 1.2.48: 8(10)
 1.2.49: (3)(11)
 1.2.50: (7)(11)
 1.2.51: 15 13 (1)
 1.2.52: 11 13 (1)
 1.2.53:  22 # 0
 1.2.54:  23 # 0
 1.2.55: 4)(2)(1)
 1.2.56: (5)(3)(2)
 1.2.57: 2(3)(1)(2)(4)
 1.2.58: 3(2)(1)(5)(3)
 1.2.59: (10) 2
 1.2.60: (8) 2
 1.2.61: 102
 1.2.62: 82
 1.2.63: (2) 3
 1.2.64: (3) 3
 1.2.65: (1) 4
 1.2.66: (4) 4
 1.2.67: (1) 33
 1.2.68: (1) 35
 1.2.69:  a 1 2 b 3
 1.2.70:  a 1 4 b 3
 1.2.71: 12 4
 1.2.72: 30 5
 1.2.73: 90 2
 1.2.74: 55 5
 1.2.75: 0 4.6
 1.2.76: 0 5.3
 1.2.77: 4.6 0
 1.2.78:  5.3 0
 1.2.79:  1 2 , a 7 9 b
 1.2.80:  1 2 , a 3 5 b
 1.2.81: 6 , a 2 5 b
 1.2.82: a 2 9 b
 1.2.83: 4(5)  6(3)
 1.2.84: 8(3)  5(6)
 1.2.85: 3(2) 2  4(3) 2
 1.2.86: 5(3) 2  2(2) 2
 1.2.87: 82  16 , 22 # 4  3
 1.2.88: 102  100 , 52 # 2  3
 1.2.89: 5 # 2  32 [32  (2)] 2
 1.2.90: 10 , 2 + 3 # 4 (12  3 # 2) 2
 1.2.91: 8  3[2(2  5)  4(8  6)]
 1.2.92: 8  3[2(5  7)  5(4  2)]
 1.2.93: 2(2)  4(3) 5  8
 1.2.94: 6(4)  5(3) 9  10
 1.2.95: (5  6) 2  2 3  7 89  3 # 52
 1.2.96: 12 , 3 # 5 22 + 32 7 + 3  62
 1.2.97: 15  23  (1) + 12 , 2 # 3
 1.2.98: 17  5  (2) + 12 , 2 # 3
 1.2.99: 20 + 1  2102  (5 + 1) 2 (2)
 1.2.100: 24 , 23 # (5  2) , [1  (3)] 2
 1.2.101: 4x + 10
 1.2.102: 5x + 30
 1.2.103: 7x
 1.2.104: 3x  7
 1.2.105: 4 + (6 + x)
 1.2.106: 12 + (3 + x)
 1.2.107: 7(3x)
 1.2.108: 10(5x)
 1.2.109:  1 3 (3y)
 1.2.110:  1 4 (4y)
 1.2.111: 3(2x + 5)
 1.2.112: 5(4x + 7)
 1.2.113: 7(2x + 3)
 1.2.114: 9(3x + 2)
 1.2.115: (3x  6)
 1.2.116: (6x  3)
 1.2.117: 7x + 5x
 1.2.118: 8x + 10x
 1.2.119: 6x2  x2
 1.2.120: 9x2  x2
 1.2.121: 6x + 10x2 + 4x + 2x2
 1.2.122: 9x + 5x2 + 3x + 4x2
 1.2.123: 8(3x  5)  6x
 1.2.124: 7(4x  5)  8x
 1.2.125: 5(3y  2)  (7y + 2)
 1.2.126: 4(5y  3)  (6y + 3)
 1.2.127: 7  4[3  (4y  5)]
 1.2.128: 6  5[8  (2y  4)]
 1.2.129: 18x2 + 4  [6(x2  2) + 5]
 1.2.130: 14x2 + 5  [7(x2  2) + 4]
 1.2.131: A number decreased by the sum of the number and four
 1.2.132: A number decreased by the difference between eight and the number
 1.2.133: Six times the product of negative five and a number
 1.2.134: Ten times the product of negative four and a number
 1.2.135: The difference between the product of five and a number and twice t...
 1.2.136: The difference between the product of six and a number and negative...
 1.2.137: The difference between eight times a number and six more than three...
 1.2.138: Eight decreased by three times the sum of a number and six
 1.2.139: What is the combined approval rating of the UK and Iran?
 1.2.140: What is the combined approval rating of Israel and China?
 1.2.141: What is the difference between the approval rating of the UK and th...
 1.2.142: What is the difference between the approval rating of Israel and th...
 1.2.143: By how much does the approval rating of France exceed the approval ...
 1.2.144: By how much does the approval rating of France exceed the approval ...
 1.2.145: What is the average of the approval ratings for China, France, and ...
 1.2.146: What is the average of the approval ratings for Iran, China, and th...
 1.2.147: According to the formula on the previous page, how much money did c...
 1.2.148: According to the formula on the previous page, how much money did c...
 1.2.149: You had $10,000 to invest. You put x dollars in a safe, government...
 1.2.150: It takes you 50 minutes to get to campus. You spend t minutes walki...
 1.2.151: What is the meaning of a in terms of a number line?
 1.2.152: Explain how to add two numbers with the same sign. Give an example ...
 1.2.153: Explain how to add two numbers with different signs. Give an exampl...
 1.2.154: What are opposites, or additive inverses? What happens when finding...
 1.2.155: Explain how to subtract real numbers.
 1.2.156: Explain how to multiply two numbers with different signs. Give an e...
 1.2.157: Explain how to multiply two numbers with the same sign.Give an exam...
 1.2.158: Explain how to determine the sign of a product that involves more t...
 1.2.159: Explain how to divide real numbers.
 1.2.160: Why is 0 4 = 0, although 4 0 is undefined?
 1.2.161: What are equivalent algebraic expressions?
 1.2.162: State a commutative property and give an example of how it is used ...
 1.2.163: State an associative property and give an example of howit is used ...
 1.2.164: State a distributive property and give an example of how it is used...
 1.2.165: What are the terms of an algebraic expression? How can you tell if ...
 1.2.166: What does it mean to simplify an algebraic expression?
 1.2.167: If a negative sign appears outside parentheses, explain how to simp...
 1.2.168: My mathematical model, although it contains an algebraic expression...
 1.2.169: Subtraction actually means the addition of an additive inverse.
 1.2.170: The terms 13x2 and 10x both contain the variable x, so I can combin...
 1.2.171: There is no number in front of the term x, so this means that the t...
 1.2.172: 16 , 4 # 2 = 16 , 8 = 2
 1.2.173: 6  2(4 + 3) = 4(4 + 3) = 4(7) = 28
 1.2.174: 5 + 3(x  4) = 8(x  4) = 8x  32
 1.2.175: x  x = x + (x) = 0
 1.2.176: x  0.02(x + 200) = 0.98x  4
 1.2.177: 8  2 # 3  4 = 14
 1.2.178: 2 # 5  1 2 # 10 # 9 = 45
 1.2.179: Simplify: 9[4  (1 + 6)]  (3  9) 2 5 + 12 5  6 2 + 1
 1.2.180: Write the following English phrase as an algebraic expression: The ...
 1.2.181: Evaluate 10 + 2(x  5)4 for x = 7. (Section 1.1, Example 3)
 1.2.182: Graph (5,) on a number line. (Section 1.1, Example 7)
 1.2.183: If y = 4  x2 , find the value of y that corresponds to values of x...
 1.2.184: If y = 1  x2 , find the value of y that corresponds to values of x...
 1.2.185: If y = x + 1 , find the value of y that corresponds tovalues of x f...
Solutions for Chapter 1.2: Operations with Real Numbers and Simplifying Algebraic Expressions
Full solutions for Intermediate Algebra for College Students  6th Edition
ISBN: 9780321758934
Solutions for Chapter 1.2: Operations with Real Numbers and Simplifying Algebraic Expressions
Get Full SolutionsChapter 1.2: Operations with Real Numbers and Simplifying Algebraic Expressions includes 185 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 185 problems in chapter 1.2: Operations with Real Numbers and Simplifying Algebraic Expressions have been answered, more than 46459 students have viewed full stepbystep solutions from this chapter. Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934. This textbook survival guide was created for the textbook: Intermediate Algebra for College Students, edition: 6.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Outer product uv T
= column times row = rank one matrix.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).