Question 1

Simplify.
-$$\frac { 4 + \frac { 2 } { x } } { \frac { x } { 4 } + \frac { 1 } { 8 } }$$

Accepted Answer

A)  $\frac { x } { 16 }$ 
B)  16 
C)  $\frac { 16 } { x }$ 
D)  1 
A)  $\frac { x } { 16 }$ 
B)  16 
C)  $\frac { 16 } { x }$ 
D)  1

Question 2

Subtract. Simplify, if possible.
-$\frac { 2 ( y - 1 ) } { 3 y - 8 } - \frac { y - 2 } { 8 - 3 y } - \frac { 3 y } { 3 y - 8 }$

Accepted Answer

A)  $\frac { - 2 y } { 3 y - 8 }$ 
B)  $\frac { 4 } { 3 y - 8 }$ 
C)  $\frac { - 4 } { ( 3 y - 8 ) ( 8 - 3 y ) }$ 
D)  $\frac { - 4 } { 3 y - 8 }$ 
A)  $\frac { - 2 y } { 3 y - 8 }$ 
B)  $\frac { 4 } { 3 y - 8 }$ 
C)  $\frac { - 4 } { ( 3 y - 8 ) ( 8 - 3 y ) }$ 
D)  $\frac { - 4 } { 3 y - 8 }$

Question 3

Find all numbers for which the rational expression is not defined.
-$\frac { x ^ { 2 } - 16 } { x ^ { 2 } - 3 x - 10 }$

Accepted Answer

A)  $x = 0$ 
B)  $x = 5$ and $x = - 2$ 
C)  $x = - 5$ and $x = 2$ 
D)  $x = 4$ and $x = - 4$ 
A)  $x = 0$ 
B)  $x = 5$ and $x = - 2$ 
C)  $x = - 5$ and $x = 2$ 
D)  $x = 4$ and $x = - 4$

Question 4

Find the reciprocal.
-$$\frac { y ^ { 2 } + 11 y + 2 } { y ^ { 2 } - y + 4 }$$

Accepted Answer

A)  $\frac { y ^ { 2 } - 2 y + 11 } { y ^ { 2 } + y + 4 }$ 
B)  $\frac { y ^ { 2 } - y - 4 } { y ^ { 2 } + 11 y - 2 }$ 
C)  $\frac { y ^ { 2 } + 11 y - 2 } { y ^ { 2 } - y - 4 }$ 
D)  $\frac { y ^ { 2 } - y + 4 } { y ^ { 2 } + 11 y + 2 }$ 
A)  $\frac { y ^ { 2 } - 2 y + 11 } { y ^ { 2 } + y + 4 }$ 
B)  $\frac { y ^ { 2 } - y - 4 } { y ^ { 2 } + 11 y - 2 }$ 
C)  $\frac { y ^ { 2 } + 11 y - 2 } { y ^ { 2 } - y - 4 }$ 
D)  $\frac { y ^ { 2 } - y + 4 } { y ^ { 2 } + 11 y + 2 }$

Question 5

Solve the equation.
-$$\frac { 3 } { x } + \frac { 7 } { 8 } = 1$$

Accepted Answer

A)  $\frac { 5 } { 4 }$ 
B)  8 
C)  10 
D)  24 
A)  $\frac { 5 } { 4 }$ 
B)  8 
C)  10 
D)  24

Question 6

Find the LCM.
-$m ^ { 2 } + 2 m , m ^ { 2 } + 7 m + 10$

Accepted Answer

A)  $m ( x - 3 ) ( m + 5 )$ 
B)  $m ( m - 3 ) ^ { 2 }$ 
C)  $( \mathrm { m } - 3 ) ^ { 2 }$ 
D)  $m ( m + 2 ) ( m + 5 )$ 
A)  $m ( x - 3 ) ( m + 5 )$ 
B)  $m ( m - 3 ) ^ { 2 }$ 
C)  $( \mathrm { m } - 3 ) ^ { 2 }$ 
D)  $m ( m + 2 ) ( m + 5 )$

Question 7

Subtract. Simplify, if possible.
-$$\frac { 2 x + 6 } { x ^ { 2 } + 6 x + 8 } - \frac { x + 4 } { x ^ { 2 } + 6 x + 8 }$$

Accepted Answer

A)  $\frac { 7 } { x ^ { 2 } + 6 x + 8 }$ 
B)  $\frac { x + 10 } { x ^ { 2 } + 6 x + 8 }$ 
C)  $x + 2$ 
D)  $\frac { 1 } { x + 4 }$ 
A)  $\frac { 7 } { x ^ { 2 } + 6 x + 8 }$ 
B)  $\frac { x + 10 } { x ^ { 2 } + 6 x + 8 }$ 
C)  $x + 2$ 
D)  $\frac { 1 } { x + 4 }$

Question 8

Add. Simplify, if possible.
-$$\frac { 9 } { 8 x ^ { 2 } } + \frac { 9 } { 8 x ^ { 2 } }$$

Accepted Answer

A)  $\frac { 4 } { 9 x ^ { 2 } }$ 
B)  9 
C)  $\frac { 9 } { 4 x ^ { 2 } }$ 
D)  $\frac { 9 } { 2 x ^ { 4 } }$ 
A)  $\frac { 4 } { 9 x ^ { 2 } }$ 
B)  9 
C)  $\frac { 9 } { 4 x ^ { 2 } }$ 
D)  $\frac { 9 } { 2 x ^ { 4 } }$

Question 9

Divide and simplify.
-$\frac { 3 x ^ { 2 } } { 5 } \div \frac { x ^ { 3 } } { 15 }$

Accepted Answer

A)  $\frac { x ^ { 5 } } { 25 }$ 
B)  $\frac { 9 x ^ { 2 } } { x ^ { 3 } }$ 
C)  $\frac { 9 } { x }$ 
D)  $\frac { x } { 9 }$ 
A)  $\frac { x ^ { 5 } } { 25 }$ 
B)  $\frac { 9 x ^ { 2 } } { x ^ { 3 } }$ 
C)  $\frac { 9 } { x }$ 
D)  $\frac { x } { 9 }$

Question 10

Answer the question.
-If 1 is substituted for $x$ in the rational expression $\frac { x - 1 } { x ^ { 2 } - 1 }$, the result is $\frac { 0 } { 0 }$. Mathematicians have been known to say "Any number divided by itself is 1 ." Does this mean that this expression is equal to 1 for $x = 1$ ? Explain why or why not.

Accepted Answer

A) True 
 B)False

Question 11

Add. Simplify, if possible.
-$$\frac { m - 4 } { m ^ { 2 } - 6 m + 5 } + \frac { 4 m - 3 } { m ^ { 2 } - 9 m + 20 }$$

Accepted Answer

A)  $\frac { 5 m ^ { 2 } - 15 m + 19 } { ( m - 5 ) ( m - 1 ) ( m - 4 ) }$ 
B)  $5 \mathrm {~m} - 7$ 
C)  $\frac { 5 m ^ { 2 } - 15 m + 19 } { ( m + 5 ) ( m + 1 ) ( m + 4 ) }$ 
D)  $\frac { 5 m - 7 } { 2 m ^ { 2 } - 15 m + 25 }$ 
A)  $\frac { 5 m ^ { 2 } - 15 m + 19 } { ( m - 5 ) ( m - 1 ) ( m - 4 ) }$ 
B)  $5 \mathrm {~m} - 7$ 
C)  $\frac { 5 m ^ { 2 } - 15 m + 19 } { ( m + 5 ) ( m + 1 ) ( m + 4 ) }$ 
D)  $\frac { 5 m - 7 } { 2 m ^ { 2 } - 15 m + 25 }$

Question 12

Solve the equation.
-$$\frac { 4 } { x - 1 } + \frac { 4 } { 2 x - 2 } = 6$$

Accepted Answer

A)  1 
B)  2 
C)  0 
D)  24 
A)  1 
B)  2 
C)  0 
D)  24

Question 13

Answer the question.
-Without multiplying by the least common denominator and solving, explain why the rational equation $\frac { x } { x - 3 } = \frac { 3 } { x - 3 }$ has no solution. (Hint: Examine both numerators and denominators carefully.)

Accepted Answer

Since the denominators are the

Question 14

Answer the question.
-If the rational expression $\frac { 7 x ^ { 3 } y ^ { 8 } } { 5 p ^ { 7 } q }$ represents the area of a rectangle and $\frac { 5 x ^ { 2 } y^6 } { p ^ { 6 } }$ represents the length of the rectangle, what rational expression represents the width?

Accepted Answer

A)  $\frac { 7 x ^ { 5 } y ^ { 14 } } { 25 p ^ { 13 } q }$ 
B)  $\frac { 7 x y^ { 2 } } { 25 p q }$ 
C)  $\frac { 7 x y ^ { 2 } } { 10 p q }$ 
D)  $\frac { 7 x ^ { 5 } y ^ { 14 } } { p ^ { 13 } q }$ 
A)  $\frac { 7 x ^ { 5 } y ^ { 14 } } { 25 p ^ { 13 } q }$ 
B)  $\frac { 7 x y^ { 2 } } { 25 p q }$ 
C)  $\frac { 7 x y ^ { 2 } } { 10 p q }$ 
D)  $\frac { 7 x ^ { 5 } y ^ { 14 } } { p ^ { 13 } q }$

Question 15

Solve.
-The distance D that a spring is stretched by a hanging object varies directly as the weight W of the object. If a 16-kg object stretches a spring 76 cm, find the distance when the weight is 19-kg. Round to the nearest Hundredths when necessary.

Accepted Answer

A) 4.75 cm 
B) 111 cm 
C) 90.25 cm 
D) 4 cm 
A) 4.75 cm 
B) 111 cm 
C) 90.25 cm 
D) 4 cm

Question 16

Solve the problem.
-To determine the number of fish in a lake, a park ranger catches 280 fish, tags them, and returns them to the lake. Later, 108 fish are caught, and it is found that 30 of them are tagged. Estimate the number of fish in the
Lake.

Accepted Answer

A) 78 
B) 12 
C) 907,200 
D) 1008 
A) 78 
B) 12 
C) 907,200 
D) 1008

Question 17

Find all numbers for which the rational expression is not defined.
-$\frac { 5 y - 3 } { y ^ { 2 } - 49 }$

Accepted Answer

A)  $y = 7$ and $y = - 7$ 
B)  $y = \frac { 3 } { 5 }$ 
C)  $y = 49$ 
D)  $y = 7$ 
A)  $y = 7$ and $y = - 7$ 
B)  $y = \frac { 3 } { 5 }$ 
C)  $y = 49$ 
D)  $y = 7$

Question 18

Divide and simplify.
-$\frac { 2 p - 2 } { p } \div \frac { 6 p - 6 } { 4 p ^ { 2 } }$

Accepted Answer

A)  $\frac { 12 p ^ { 2 } + 24 p + 12 } { 4 p ^ { 3 } }$ 
B)  $\frac { 3 } { 4 p }$ 
C)  $\frac { 4 p } { 3 }$ 
D)  $\frac { 8 p ^ { 3 } - 8 p ^ { 2 } } { 6 p ^ { 2 } - 6 p }$ 
A)  $\frac { 12 p ^ { 2 } + 24 p + 12 } { 4 p ^ { 3 } }$ 
B)  $\frac { 3 } { 4 p }$ 
C)  $\frac { 4 p } { 3 }$ 
D)  $\frac { 8 p ^ { 3 } - 8 p ^ { 2 } } { 6 p ^ { 2 } - 6 p }$

Question 19

Answer the question.
-Explain in your own words how to multiply rational expressions. (How would you explain this to a student in this class who was absent from class?).

Accepted Answer

To multiply rational expressions, you fi

Question 20

Find all numbers for which the rational expression is not defined.
-$$\frac { c - 2 } { 3 }$$

Accepted Answer

A) c = 2 
B) None 
C) c = 0 
D) c = -2 
A) c = 2 
B) None 
C) c = 0 
D) c = -2

Simplify. - $\frac { 4 + \frac { 2 } { x } } { \frac { x } { 4 } + \frac { 1 } { 8 } }$

Subtract. Simplify, if possible. - $\frac { 2 ( y - 1 ) } { 3 y - 8 } - \frac { y - 2 } { 8 - 3 y } - \frac { 3 y } { 3 y - 8 }$

Find all numbers for which the rational expression is not defined. - $\frac { x ^ { 2 } - 16 } { x ^ { 2 } - 3 x - 10 }$

Find the reciprocal. - $\frac { y ^ { 2 } + 11 y + 2 } { y ^ { 2 } - y + 4 }$

Solve the equation. - $\frac { 3 } { x } + \frac { 7 } { 8 } = 1$

Find the LCM. - $m ^ { 2 } + 2 m , m ^ { 2 } + 7 m + 10$

Subtract. Simplify, if possible. - $\frac { 2 x + 6 } { x ^ { 2 } + 6 x + 8 } - \frac { x + 4 } { x ^ { 2 } + 6 x + 8 }$

Add. Simplify, if possible. - $\frac { 9 } { 8 x ^ { 2 } } + \frac { 9 } { 8 x ^ { 2 } }$

Divide and simplify. - $\frac { 3 x ^ { 2 } } { 5 } \div \frac { x ^ { 3 } } { 15 }$

Add. Simplify, if possible. - $\frac { m - 4 } { m ^ { 2 } - 6 m + 5 } + \frac { 4 m - 3 } { m ^ { 2 } - 9 m + 20 }$

Solve the equation. - $\frac { 4 } { x - 1 } + \frac { 4 } { 2 x - 2 } = 6$

Answer the question. -Without multiplying by the least common denominator and solving, explain why the rational equation $\frac { x } { x - 3 } = \frac { 3 } { x - 3 }$ has no solution. (Hint: Examine both numerators and denominators carefully.)

Answer the question. -If the rational expression $\frac { 7 x ^ { 3 } y ^ { 8 } } { 5 p ^ { 7 } q }$ represents the area of a rectangle and $\frac { 5 x ^ { 2 } y^6 } { p ^ { 6 } }$ represents the length of the rectangle, what rational expression represents the width?

Solve. -The distance D that a spring is stretched by a hanging object varies directly as the weight W of the object. If a 16-kg object stretches a spring 76 cm, find the distance when the weight is 19-kg. Round to the nearest Hundredths when necessary.

Solve the problem. -To determine the number of fish in a lake, a park ranger catches 280 fish, tags them, and returns them to the lake. Later, 108 fish are caught, and it is found that 30 of them are tagged. Estimate the number of fish in the Lake.

Find all numbers for which the rational expression is not defined. - $\frac { 5 y - 3 } { y ^ { 2 } - 49 }$

Divide and simplify. - $\frac { 2 p - 2 } { p } \div \frac { 6 p - 6 } { 4 p ^ { 2 } }$

Answer the question. -Explain in your own words how to multiply rational expressions. (How would you explain this to a student in this class who was absent from class?).

Find all numbers for which the rational expression is not defined. - $\frac { c - 2 } { 3 }$

Introduction to Real Numbers and Algebraic Expressions

Solving Equations and Inequalities

Graphs of Linear Equations

Polynomials: Operations

Polynomials: Factoring

Graphs, Functions, and Applications

Systems of Equations

More on Inequalities

Radical Expressions, Equations, and Functions

Quadratic Equations and Functions

Exponential Functions and Logarithmic Functions

Filters

Exam 6: Rational Expressions and Equations

Simplify. - 4+2xx4+18\frac { 4 + \frac { 2 } { x } } { \frac { x } { 4 } + \frac { 1 } { 8 } }4x​+81​4+x2​​

Subtract. Simplify, if possible. - 2(y−1)3y−8−y−28−3y−3y3y−8\frac { 2 ( y - 1 ) } { 3 y - 8 } - \frac { y - 2 } { 8 - 3 y } - \frac { 3 y } { 3 y - 8 }3y−82(y−1)​−8−3yy−2​−3y−83y​

Find all numbers for which the rational expression is not defined. - x2−16x2−3x−10\frac { x ^ { 2 } - 16 } { x ^ { 2 } - 3 x - 10 }x2−3x−10x2−16​

Find the reciprocal. - y2+11y+2y2−y+4\frac { y ^ { 2 } + 11 y + 2 } { y ^ { 2 } - y + 4 }y2−y+4y2+11y+2​

Solve the equation. - 3x+78=1\frac { 3 } { x } + \frac { 7 } { 8 } = 1x3​+87​=1

Find the LCM. - m2+2m,m2+7m+10m ^ { 2 } + 2 m , m ^ { 2 } + 7 m + 10m2+2m,m2+7m+10

Subtract. Simplify, if possible. - 2x+6x2+6x+8−x+4x2+6x+8\frac { 2 x + 6 } { x ^ { 2 } + 6 x + 8 } - \frac { x + 4 } { x ^ { 2 } + 6 x + 8 }x2+6x+82x+6​−x2+6x+8x+4​

Add. Simplify, if possible. - 98x2+98x2\frac { 9 } { 8 x ^ { 2 } } + \frac { 9 } { 8 x ^ { 2 } }8x29​+8x29​

Divide and simplify. - 3x25÷x315\frac { 3 x ^ { 2 } } { 5 } \div \frac { x ^ { 3 } } { 15 }53x2​÷15x3​

Add. Simplify, if possible. - m−4m2−6m+5+4m−3m2−9m+20\frac { m - 4 } { m ^ { 2 } - 6 m + 5 } + \frac { 4 m - 3 } { m ^ { 2 } - 9 m + 20 }m2−6m+5m−4​+m2−9m+204m−3​

Solve the equation. - 4x−1+42x−2=6\frac { 4 } { x - 1 } + \frac { 4 } { 2 x - 2 } = 6x−14​+2x−24​=6

Answer the question. -Without multiplying by the least common denominator and solving, explain why the rational equation xx−3=3x−3\frac { x } { x - 3 } = \frac { 3 } { x - 3 }x−3x​=x−33​ has no solution. (Hint: Examine both numerators and denominators carefully.)