 6.Cumulative Review: Translate each phrase to an algebraic expression. Use thevariable x...
 6.Review: (6.1) Find the domain for each rational function.f 1x2 = 3  5x7
 6.Test: Find the domain of each rational function f 1x2 = 5x21  x
 6.Cumulative Review: Translate each phrase to an algebraic expression. Use thevariable x...
 6.Review: (6.1) Find the domain for each rational function.g1x2 = 2x + 411
 6.Test: Find the domain of each rational functiong1x2 = 9x2  9x2 + 4x + 3
 6.Cumulative Review: Solve for y:y3  y4 = 16
 6.Review: (6.1) Find the domain for each rational function.F1x2 = 3x2x  5
 6.Test: Write each rational expression in lowest terms.7x  2124  8x
 6.Cumulative Review: Solve x7+ x5 = 125 .
 6.Review: (6.1) Find the domain for each rational function.h1x2 = 4x3x  12
 6.Test: Write each rational expression in lowest terms.x2  4xx2 + 5x  36
 6.Cumulative Review: In the United States, the annual consumption of cigarettesis declin...
 6.Review: (6.1) Find the domain for each rational function.f 1x2 = x3 + 2x2 + 8x
 6.Test: Write each rational expression in lowest terms.x3  8x  2
 6.Cumulative Review: Olivia has scores of 78, 65, 82, and 79 on her algebra tests. Usean...
 6.Review: (6.1) Find the domain for each rational function.G1x2 = 203x2  48
 6.Test: Perform the indicated operation. If possible, simplify your answer....
 6.Cumulative Review: Solve: `3x + 12 ` = 2
 6.Review: Write each rational expression in lowest terms.x  1212  x
 6.Test: Perform the indicated operation. If possible, simplify your answer....
 6.Cumulative Review: Solve: `2x  13 ` + 6 = 3
 6.Review: Write each rational expression in lowest terms.5x  1525x  75
 6.Test: Perform the indicated operation. If possible, simplify your answer....
 6.Cumulative Review: Solve for x: `21x + 123 ` 0
 6.Review: Write each rational expression in lowest terms.2x2x2  2x
 6.Test: Perform the indicated operation. If possible, simplify your answer....
 6.Cumulative Review: Solve for x: `31x  124 ` 2
 6.Review: Write each rational expression in lowest terms.x + 7x2  49
 6.Test: Perform the indicated operation. If possible, simplify your answer....
 6.Cumulative Review: Graph the equation y = 2x + 3.
 6.Review: Write each rational expression in lowest terms. 2x2 + 4x  30x2 + x...
 6.Test: Perform the indicated operation. If possible, simplify your answer....
 6.Cumulative Review: Graph the equation y = x + 3.
 6.Review: The average cost (per bookcase) of manufacturing x bookcasesis give...
 6.Test: Perform the indicated operation. If possible, simplify your answer....
 6.Cumulative Review: Determine whether the following relations are also functions.a. 51 ...
 6.Review: Perform each indicated operation. Write your answers in lowest term...
 6.Test: Perform the indicated operation. If possible, simplify your answer....
 6.Cumulative Review: If f 1x2 = x2 + 3x  2, finda. f(0) b. f 1 32 c. f a13b
 6.Review: Perform each indicated operation. Write your answers in lowest term...
 6.Test: Perform the indicated operation. If possible, simplify your answer....
 6.Cumulative Review: Find the intercepts and graph: 3x + 4y = 12.
 6.Review: Perform each indicated operation. Write your answers in lowest term...
 6.Test: Simplify each complex fraction.5x  73x98x  1x
 6.Cumulative Review: Find the intercepts and graph: 3x  y = 6.
 6.Review: Perform each indicated operation. Write your answers in lowest term...
 6.Test: Simplify each complex fraction.x2  5x + 6x + 3x2  4x + 4x2  9
 6.Cumulative Review: Find an equation of the line with slope 3 containing the point 11,...
 6.Review: Perform each indicated operation. Write your answers in lowest term...
 6.Test: Divide.14x2y + 9x + 3xz2 , 3xz
 6.Cumulative Review: Find an equation of the line with slope 1 2 containing the point 1 ...
 6.Review: Perform each indicated operation. Write your answers in lowest term...
 6.Test: Divide.14x3  5x2 , 12x + 12
 6.Cumulative Review: Graph the intersection of x 1 and y 2x  1.
 6.Review: Perform each indicated operation. Write your answers in lowest term...
 6.Test: Use synthetic division to divide 14x4  3x3  x  12 by1x + 32 .
 6.Cumulative Review: Graph the union of 2x + y 4 or y 7 2.
 6.Review: Perform each indicated operation. Write your answers in lowest term...
 6.Test: If P1x2 = 4x4 + 7x2  2x  5, use the remainder theoremto find P1 ...
 6.Cumulative Review: Use the elimination method to solve the system.e3x  2y = 104x  3y...
 6.Review: Perform each indicated operation. Write your answers in lowest term...
 6.Test: Solve each equation for x.5x + 33x  7 = 197
 6.Test: Solve each equation for x.xx  4 = 3  4x  4
 6.Cumulative Review: Use the substitution method to solve the system.e 2x + 3y = 63x  ...
 6.Review: Perform each indicated operation. Write your answers in lowest term...
 6.Test: Solve each equation for x.3x + 2  15x = 25x2 + 10x
 6.Cumulative Review: Solve the system. 2x  4y + 8z = 2x  3y + z = 11x  2y + 4z = 0
 6.Review: (6.2) Find the LCD of the rational expressions in each list54x2y5, ...
 6.Test: Solve each equation for x.x2 + 8x  1 = 21x + 42x
 6.Cumulative Review: Solve the system. 2x  2y + 4z = 64x  y + z = 83x  y + z = 6
 6.Review: (6.2) Find the LCD of the rational expressions in each list52x, 7x  2
 6.Test: Solve for x: x + b a = 4x  7a b
 6.Cumulative Review: The measure of the largest angle of a triangle is 80 more than the ...
 6.Review: (6.2) Find the LCD of the rational expressions in each list35x, 2x  5
 6.Test: The product of one more than a number and twice the reciprocal of t...
 6.Cumulative Review: Kernersville office supply sold three reams of paper and two boxes ...
 6.Review: (6.2) Find the LCD of the rational expressions in each list15x3, 4x...
 6.Test: If Jan can weed the garden in 2 hours and her husband can weed it i...
 6.Cumulative Review: Use matrices to solve the system. x + 2y + z = 22x  y + 2z = 5x +...
 6.Review: Perform each indicated operation. Write your answers in lowest term...
 6.Test: Suppose that W is inversely proportional to V. If W = 20 when V = 1...
 6.Cumulative Review: Use matrices to solve the system. x + y + z = 92x  2y + 3z = 23x ...
 6.Review: Perform each indicated operation. Write your answers in lowest term...
 6.Test: Suppose that Q is jointly proportional to R and the square of S. If...
 6.Cumulative Review: Evaluate the following.a. 70 b. 70c. 12x + 520 d. 2x0
 6.Review: Perform each indicated operation. Write your answers in lowest term...
 6.Test: When an anvil is dropped into a gorge, the speed withwhich it strik...
 6.Cumulative Review: Simplify the following. Write answers with positive exponents.a. 2...
 6.Review: Perform each indicated operation. Write your answers in lowest term...
 6.Cumulative Review: Simplify each expression. Assume that a and b are integersand that ...
 6.Review: Perform each indicated operation. Write your answers in lowest term...
 6.Cumulative Review: Simplify each expression. Assume that a and b are integersand that ...
 6.Review: Perform each indicated operation. Write your answers in lowest term...
 6.Cumulative Review: Find the degree of each term.a. 3x2 b. 23x5 c. yd. 12x2yz3 e. 5.27
 6.Review: Perform each indicated operation. Write your answers in lowest term...
 6.Cumulative Review: Subtract 12x  72 from 2x2 + 8x  3
 6.Review: Perform each indicated operation. Write your answers in lowest term...
 6.Cumulative Review: Multiply 33 + 12a + b242 .
 6.Review: Perform each indicated operation. Write your answers in lowest term...
 6.Cumulative Review: Multiply 34 + 13x  y242 .
 6.Review: Perform each indicated operation. Write your answers in lowest term...
 6.Cumulative Review: Factor ab  6a + 2b  12.
 6.Review: (6.3) Simplify each complex fraction.1  3x42 + x4
 6.Cumulative Review: Factor xy + 2x  5y  10.
 6.Review: (6.3) Simplify each complex fraction.x215x + 15x
 6.Cumulative Review: Factor 2n2  38n + 80.
 6.Review: (6.3) Simplify each complex fraction.2  32xx  25x
 6.Cumulative Review: Factor 6x2  x  35
 6.Review: (6.3) Simplify each complex fraction.1 + xyx2y2  1
 6.Cumulative Review: Factor x2 + 4x + 4  y2 .
 6.Review: (6.3) Simplify each complex fraction.5x +1xy3x2
 6.Review: (6.3) Simplify each complex fraction.x3  3x1 +3x
 6.Cumulative Review: Factor 4x2  4x + 1  9y2 .
 6.Review: (6.3) Simplify each complex fraction.1x  1+ 11x + 1  1
 6.Cumulative Review: Solve 1x + 221x  62 = 0.
 6.Review: (6.3) Simplify each complex fraction.x  3x + 3+ x + 3x  3x  3x +...
 6.Cumulative Review: Solve 2x13x + 121x  32 = 0.
 6.Review: If f 1x2 = 3 x , x 0, find each of the following.f 1a + h2
 6.Cumulative Review: Simplify each rational expression.a. 2x210x3  2x2b. 9x2 + 13x + 48...
 6.Review: If f 1x2 = 3 x , x 0, find each of the following.f 1a2
 6.Cumulative Review: For the graph of f(x), answer the following:a. Find the domain and ...
 6.Review: If f 1x2 = 3 x , x 0, find each of the following.Use Exercises 45 a...
 6.Cumulative Review: Subtract 5kk2  4  2k2 + k  2
 6.Review: If f 1x2 = 3 x , x 0, find each of the following.Simplify the resul...
 6.Cumulative Review: Subtract 5aa2  4  32  a
 6.Review: 14xy + 2x2  92 , 4xy
 6.Cumulative Review: Solve 3x  x + 213x = 53
 6.Review: Divide 12xb2 + 16xb4 by 4xb3.
 6.Cumulative Review: Solve 3x  x + 213x = 53
 6.Review: 13x4  25x2  202 , 1x  32
 6.Review: 1 x2 + 2x4 + 5x  122 , 1x + 22
 6.Review: 12x3 + 3x2  2x + 22 , 12x + 32
 6.Review: 13x4 + 5x3 + 7x2 + 3x  22 , 1x2 + x + 22
 6.Review: Use synthetic division to find each quotient.13x3 + 12x  42 , 1x  22
 6.Review: Use synthetic division to find each quotient.1x5  12 , 1x + 12
 6.Review: Use synthetic division to find each quotient.1x3  812 , 1x  32
 6.Review: Use synthetic division to find each quotient.13x4  2x2 + 102 , 1x ...
 6.Review: If P1x2 = 3x5  9x + 7, use the remainder theorem to find the follo...
 6.Review: If P1x2 = 3x5  9x + 7, use the remainder theorem to find the follo...
 6.Review: If P1x2 = 3x5  9x + 7, use the remainder theorem to find the follo...
 6.Review: If the area of the rectangle is 1x4  x3  6x2  6x + 182square mil...
 6.Review: Solve each equation. 3x +13 = 5x
 6.Review: Solve each equation.2x + 35x  9 = 32
 6.Review: Solve each equation.1x  2  3xx2  4 = 2x + 2
 6.Review: Solve each equation.7x  x7 = 0
 6.Review: Solve each equation or perform each indicated operation. Simplify.5...
 6.Review: Solve each equation or perform each indicated operation. Simplify.4...
 6.Review: Solve each equation or perform each indicated operation. Simplify.3...
 6.Review: Solve each equation or perform each indicated operation. Simplify.2...
 6.Review: (6.6) Solve each equation for the specified variable.A = h1a + b22 ...
 6.Review: (6.6) Solve each equation for the specified variable.1R = 1R1+1R2 f...
 6.Review: (6.6) Solve each equation for the specified variable.I = ER + r for R
 6.Review: (6.6) Solve each equation for the specified variable.A = P + Prt for r
 6.Review: (6.6) Solve each equation for the specified variable.1x = 1y  1z f...
 6.Review: (6.6) Solve each equation for the specified variable.H = kA1T1  T2...
 6.Review: Solve.The sum of a number and twice its reciprocal is 3. Find thenu...
 6.Review: Solve.If a number is added to the numerator of 37, and twice thatnu...
 6.Review: Solve.Three boys can paint a fence in 4 hours, 5 hours, and 6 hours...
 6.Review: Solve.If Sue Katz can type a certain number of mailing labels in6 h...
 6.Review: Solve.The speed of a Ranger boat in still water is 32 mph. If the b...
 6.Review: Solve.The speed of a jogger is 3 mph faster than the speed of a wal...
 6.Review: (6.7) Solve each variation problemA is directly proportional to B. ...
 6.Review: (6.7) Solve each variation problemAccording to Boyles law, the pres...
 6.Review: For expressions, perform the indicated operation and/or simplify. F...
 6.Review: For expressions, perform the indicated operation and/or simplify. F...
 6.Review: For expressions, perform the indicated operation and/or simplify. F...
 6.Review: For expressions, perform the indicated operation and/or simplify. F...
 6.Review: For expressions, perform the indicated operation and/or simplify. F...
 6.Review: For expressions, perform the indicated operation and/or simplify. F...
 6.Review: For expressions, perform the indicated operation and/or simplify. F...
 6.Review: For expressions, perform the indicated operation and/or simplify. F...
 6.Review: For expressions, perform the indicated operation and/or simplify. F...
 6.Review: For expressions, perform the indicated operation and/or simplify. F...
 6.Review: For expressions, perform the indicated operation and/or simplify. F...
 6.Review: For expressions, perform the indicated operation and/or simplify. F...
 6.Review: For expressions, perform the indicated operation and/or simplify. F...
 6.Review: For expressions, perform the indicated operation and/or simplify. F...
 6.Review: For expressions, perform the indicated operation and/or simplify. F...
 6.Review: For expressions, perform the indicated operation and/or simplify. F...
 6.Review: The denominator of a fraction is 2 more than the numerator. If the ...
 6.Review: The sum of the reciprocals of two consecutive even integers is  9 ...
 6.Review: The inlet pipe of a water tank can fill the tank in 2 hours and 30 ...
 6.Review: Timmy Garnica drove 210 miles in the same amount of time that it to...
 6.Review: Two Amtrak trains traveling on parallel tracks leave Tucson at the ...
 6.Review: The surface area of a sphere varies directly as the square of its r...
 6.Review: In 2013, it is predicted that 3 out of 4 U.S. citizens will own MP3...
 6.Review: Currently, 8 out of 25 U.S. Internet users purchase digital music. ...
 6.Review: Divide 1x3  x2 + 3x4  22 by 1x  42.
 6.Review: C is inversely proportional to D. If C = 12 when D = 8, find C when...
Solutions for Chapter 6: Rational Expressions
Full solutions for Intermediate Algebra  6th Edition
ISBN: 9780321785046
Solutions for Chapter 6: Rational Expressions
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Intermediate Algebra was written by and is associated to the ISBN: 9780321785046. This textbook survival guide was created for the textbook: Intermediate Algebra, edition: 6. Since 190 problems in chapter 6: Rational Expressions have been answered, more than 59790 students have viewed full stepbystep solutions from this chapter. Chapter 6: Rational Expressions includes 190 full stepbystep solutions.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.

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