Question 1

Sketch the graph of the inequality below. $x < - 3$

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 2

Find which of the following system of inequalities has a right triangle as a graphed solution set. System I: $\left\{ \begin{array} { l } 
y \leq \frac { 1 } { 5 } x + 4 \
y \geq 0 \
x \geq 0
\end{array} \right.$ System II: $\left\{ \begin{array} { l } 
y \leq \frac { 4 } { 5 } x - 4 \
y \geq 0 \
x \geq 0
\end{array} \right.$ System III: $\left\{ \begin{array} { l } 
y \leq \frac { 4 } { 5 } x + 5 \
y \geq 0 \
x \geq 0
\end{array} \right.$

Accepted Answer

A) Only systems I and III have a right triangle as a graphed solution set. 
B) Only systems II and III have a right triangle as a graphed solution set. 
C) No systems have a right triangle as a graphed solution set. 
D) Only system III has a right triangle as a graphed solution set. 
E) Only systems I and II have a right triangle as a graphed solution set. 
A) Only systems I and III have a right triangle as a graphed solution set. 
B) Only systems II and III have a right triangle as a graphed solution set. 
C) No systems have a right triangle as a graphed solution set. 
D) Only system III has a right triangle as a graphed solution set. 
E) Only systems I and II have a right triangle as a graphed solution set.

Question 3

An investor has $750,000 to invest in two types of investments. Type A pays 4% annually and type B pays 5% annually. To have a well-balanced portfolio, the investor imposes the following conditions. At least one-third of the total portfolio is to be allocated to type A investments and at least one-third of the portfolio is to be allocated to type B investments. What is the optimal amount that should be invested in each investment?

Accepted Answer

A) $250,000 in type A (4%), $500,000 in type B (5%) 
B) $0 in type A (4%), $750,000 in type B (5%) 
C) $500,000 in type A (4%), $250,000 in type B (5%) 
D) $750,000 in type A (4%), $0 in type B (5%) 
E) $260,000 in type A (4%), $490,000 in type B (5%) 
A) $250,000 in type A (4%), $500,000 in type B (5%) 
B) $0 in type A (4%), $750,000 in type B (5%) 
C) $500,000 in type A (4%), $250,000 in type B (5%) 
D) $750,000 in type A (4%), $0 in type B (5%) 
E) $260,000 in type A (4%), $490,000 in type B (5%)

Question 4

Find the minimum and maximum values of the objective function $z = 2 x + 8 y$ and where they occur, subject to the constraints $$x \geq 0 , y \geq 0,2 x + y \leq 4$$ .

Accepted Answer

A) The minimum value is 0 at $( 0,0 )$ and the maximum value is 32 at $( 0,4 )$ . 
B) The minimum value is 0 at $( 0,2 )$ and the maximum value is 32 at $( 0,4 )$ . 
C) The minimum value is 0 at $( 0,0 )$ and the maximum value is 32 at $( 0,2 )$ . 
D) The minimum value is 0 at $( 0,4 )$ and the maximum value is 32 at $( 0,2 )$ . 
E) The minimum value is 0 at $( 0,4 )$ and the maximum value is 32 at $( 0,0 )$ . 
A) The minimum value is 0 at $( 0,0 )$ and the maximum value is 32 at $( 0,4 )$ . 
B) The minimum value is 0 at $( 0,2 )$ and the maximum value is 32 at $( 0,4 )$ . 
C) The minimum value is 0 at $( 0,0 )$ and the maximum value is 32 at $( 0,2 )$ . 
D) The minimum value is 0 at $( 0,4 )$ and the maximum value is 32 at $( 0,2 )$ . 
E) The minimum value is 0 at $( 0,4 )$ and the maximum value is 32 at $( 0,0 )$ .

Question 5

For the given supply and demand equations, find the consumer surplus. Round to the nearest dollar. Demand Supply
P = 160 - 0.00006x p = 110 + 0.00003x

Accepted Answer

A) $9,259,259 
B) $10,185,185 
C) $11,111,111 
D) $13,888,889 
E) $4,629,630 
A) $9,259,259 
B) $10,185,185 
C) $11,111,111 
D) $13,888,889 
E) $4,629,630

Question 6

Maximize the object function $z = 4 x + 3 y$ subject to the constraints $3 x + y \leq 15,4 x + 3 y \leq 30 , x \geq 0 , \text { and } y \geq 0$ .

Accepted Answer

A) The maximum value is 24 along the line segment connection $( 0,10 ) \text { and } ( 3,4 )$ . 
B) The maximum value is 30 along the line segment connection $( 0,15 ) \text { and } ( 3,6 )$ . 
C) The maximum value is 30 along the line segment connection $( 0,10 ) \text { and } ( 3,6 )$ . 
D) The maximum value is 24 along the line segment connection $( 0,15 ) \text { and } ( 3,4 )$ . 
E) The maximum value is 30 along the line segment connection $( 0,30 ) \text { and } ( 3,6 )$ . 
A) The maximum value is 24 along the line segment connection $( 0,10 ) \text { and } ( 3,4 )$ . 
B) The maximum value is 30 along the line segment connection $( 0,15 ) \text { and } ( 3,6 )$ . 
C) The maximum value is 30 along the line segment connection $( 0,10 ) \text { and } ( 3,6 )$ . 
D) The maximum value is 24 along the line segment connection $( 0,15 ) \text { and } ( 3,4 )$ . 
E) The maximum value is 30 along the line segment connection $( 0,30 ) \text { and } ( 3,6 )$ .

Question 7

Sketch the graph of the solution set of each system of inequalities. $2 x - 5 y < - 6$ $3 x + y < 8$

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 8

An accounting firm charges $2500 for an audit and $350 for a tax return. Research and available resources have indicated the following constraints. The firm has 900 hours of staff time available each week.
The firm has 155 hours of review time available each week.
Each audit requires 75 hours of staff time and 10 hours of review time.
Each tax return requires 12.5 hours of staff time and 2.5 hours of review time.
What numbers of audits and tax returns will bring in an optimal revenue?

Accepted Answer

A) The optimal revenue of $0 will occurs when 0 audits and no tax returns are processed. 
B) The optimal revenue of $27,200 will occurs when 5 audits and no tax returns are processed. 
C) The optimal revenue of $21,700 will occurs when 62 audits and no tax returns are processed. 
D) The optimal revenue of $30,000 will occurs when 12 audits and no tax returns are processed. 
E) The optimal revenue of $27,200 will occurs when 42 audits and no tax returns are processed. 
A) The optimal revenue of $0 will occurs when 0 audits and no tax returns are processed. 
B) The optimal revenue of $27,200 will occurs when 5 audits and no tax returns are processed. 
C) The optimal revenue of $21,700 will occurs when 62 audits and no tax returns are processed. 
D) The optimal revenue of $30,000 will occurs when 12 audits and no tax returns are processed. 
E) The optimal revenue of $27,200 will occurs when 42 audits and no tax returns are processed.

Question 9

A total of $50,000 is invested in two funds paying 8% and 13% simple interest. The total yearly interest is $5100. How much is invested at the 8% rate?

Accepted Answer

A) $22,000 
B) $26,000 
C) $19,000 
D) $28,000 
E) $30,000 
A) $22,000 
B) $26,000 
C) $19,000 
D) $28,000 
E) $30,000

Question 10

Which of the following vertices of the constraint region shown is a maximum value of the objective function below.   $z = - 2 x - y - 5$

Accepted Answer

A) vertices A and D 
B) only vertex C 
C) vertices A and B 
D) vertices C and D 
E) only vertex A 
A) vertices A and D 
B) only vertex C 
C) vertices A and B 
D) vertices C and D 
E) only vertex A

Question 11

Which of the following vertices of the constraint region shown is a minimum value of the objective function below.   $$z = - 6 x - 4 y + 2$$

Accepted Answer

A) only vertex B 
B) vertices B and C 
C) vertices A and B 
D) only vertex D 
E) only vertex C 
A) only vertex B 
B) vertices B and C 
C) vertices A and B 
D) only vertex D 
E) only vertex C

Question 12

Use back-substitution to solve the system of linear equations. $$\left\{ \begin{array} { r } 
- x + 7 y - 5 z = 51 \
5 y - 8 z = 82 \
z = - 9
\end{array} \right.$$

Accepted Answer

A)  $( 2,8 , - 9 )$ 
B)  $( 8,2 , - 9 )$ 
C)  $( 8 , - 9,2 )$ 
D)  $( - 9,2,8 )$ 
E)  $( 0,2 , - 9 )$ 
A)  $( 2,8 , - 9 )$ 
B)  $( 8,2 , - 9 )$ 
C)  $( 8 , - 9,2 )$ 
D)  $( - 9,2,8 )$ 
E)  $( 0,2 , - 9 )$

Question 13

You are offered two different jobs. Company A offers an annual salary of $32,500 plus a year-end bonus of 2.5% of your total sales. Company B offers a salary of $28,000 plus a year-end bonus of 5.5% of your total sales. What is the amount you must sell in one year to earn the same salary working for either company?

Accepted Answer

A) $130,000 
B) $170,000 
C) $150,000 
D) $110,000 
E) $120,000 
A) $130,000 
B) $170,000 
C) $150,000 
D) $110,000 
E) $120,000

Question 14

Sketch the graph of the solution set of each system of inequalities. $y > 2 x - 1$ $y > x ^ { 2 } - 2 x + 2$

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 15

Solve the system by the method of elimination. $$\left\{ \begin{array} { l } 
0.02 x - 0.02 y = 0.17 \
0.01 x + 0.05 y = 0.18
\end{array} \right.$$

Accepted Answer

A)  $\left( 7 , - \frac { 3 } { 2 } \right)$ 
B)  $\left( 2 , \frac { 16 } { 5 } \right)$ 
C)  $\left( \frac { 121 } { 12 } , \frac { 19 } { 12 } \right)$ 
D)  $\left( a , \frac { - 17 + 2 a } { 2 } \right) \text { (dependent) }$ 
E) inconsistent 
A)  $\left( 7 , - \frac { 3 } { 2 } \right)$ 
B)  $\left( 2 , \frac { 16 } { 5 } \right)$ 
C)  $\left( \frac { 121 } { 12 } , \frac { 19 } { 12 } \right)$ 
D)  $\left( a , \frac { - 17 + 2 a } { 2 } \right) \text { (dependent) }$ 
E) inconsistent

Question 16

Solve the system by the method of substitution. $\left\{ \begin{aligned}
x ^ { 2 } + y ^ { 2 } & = 625 \
7 x - 24 y & = 0
\end{aligned} \right.$

Accepted Answer

A) (24, 7), (-24, -7) 
B) (24, 7) 
C) (24, 7), (-24, 7), (-24, -7), (24, -7) 
D) (7, 24), (7, -24) 
E) no real solution 
A) (24, 7), (-24, -7) 
B) (24, 7) 
C) (24, 7), (-24, 7), (-24, -7), (24, -7) 
D) (7, 24), (7, -24) 
E) no real solution

Question 17

Solve the system of linear equations. $\left\{ \begin{aligned}
- 3 x + 2 y - z & = 9 \
- x - 3 y + z & = 12 \
x - 2 y + 4 z & = 13
\end{aligned} \right.$

Accepted Answer

A)  $( - 6 , - 4,5 )$ 
B)  $( - 5 , - 1,4 )$ 
C)  $( - 1,4 , - 5 )$ 
D)  $( - 4,5 , - 6 )$ 
E) inconsistent 
A)  $( - 6 , - 4,5 )$ 
B)  $( - 5 , - 1,4 )$ 
C)  $( - 1,4 , - 5 )$ 
D)  $( - 4,5 , - 6 )$ 
E) inconsistent

Question 18

Determine which ordered pair is a solution of the system. $$\left\{ \begin{array} { l } 
3 x - 4 y = 8 \
4 x - 7 y = 19
\end{array} \right.$$

Accepted Answer

A) (-5, -4) 
B) (4, -5) 
C) (2, 7) 
D) (-4, -5) 
E) (-7, 2) 
A) (-5, -4) 
B) (4, -5) 
C) (2, 7) 
D) (-4, -5) 
E) (-7, 2)

Question 19

A company makes two types of telephone answering machines: the standard model and the deluxe model. Each machine passes through three processes: $P _ { 1 } ,$ $P _ { 2 }$ and $P _ { 3 }.$ One standard answering machine requires 1 hour in $P _ { 1 } ,$ 4 hours in $P _ { 2 },$ and 3 hours in $P _ { 3 }.$ One deluxe answering machine requires 3 hours in $P _ { 1 } ,$ 5 hours in $P _ { 2 },$ and 1 hour in $P _ { 3 }.$ Because of employee work schedules, $P _ { 1 }$ is available for 24 hours, $P _ { 2 }$ is available for 47 hours, and $P _ { 3 }$ is available for 27 hours. If the profit is $22 for each standard model and $23 for each deluxe model, how many units of each type should the company produce to maximize profit?

Accepted Answer

A) 8 standard models and 3 deluxe models 
B) 9 standard models and 0 deluxe models 
C) 8 standard models and 0 deluxe models 
D) 0 standard models and 8 deluxe models 
E) 3 standard models and 7 deluxe models 
A) 8 standard models and 3 deluxe models 
B) 9 standard models and 0 deluxe models 
C) 8 standard models and 0 deluxe models 
D) 0 standard models and 8 deluxe models 
E) 3 standard models and 7 deluxe models

Question 20

An ice cream supplier has two machines that produce vanilla and chocolate ice cream. To meet one of its contractual obligations, the company must produce at least 9 gallons of vanilla ice cream and 15 gallons of chocolate ice cream per day. One machine makes 6 gallons of vanilla and 3 gallons of chocolate ice cream per hour. The second machine makes 1 gallons of vanilla and 4 gallons of chocolate ice cream per hour. It costs $28 per hour to run machine 1 and $22 per hour to run machine 2. How many hours should each machine be operated to fulfill the contract at the least expense?

Accepted Answer

A) Machine 1 for 9 hours and machine 2 for no hours. 
B) Machine 1 for 3 hours and machine 2 for 1 hour. 
C) Machine 1 for 1 hour and machine 2 for 3 hours. 
D) Machine 1 for no hours and machine 2 for 1 hour. 
E) Machine 1 for no hours and machine 2 for 3 hours. 
A) Machine 1 for 9 hours and machine 2 for no hours. 
B) Machine 1 for 3 hours and machine 2 for 1 hour. 
C) Machine 1 for 1 hour and machine 2 for 3 hours. 
D) Machine 1 for no hours and machine 2 for 1 hour. 
E) Machine 1 for no hours and machine 2 for 3 hours.

Sketch the graph of the inequality below. $x < - 3$

For the given supply and demand equations, find the consumer surplus. Round to the nearest dollar. Demand Supply P = 160 - 0.00006x p = 110 + 0.00003x

Maximize the object function $z = 4 x + 3 y$ subject to the constraints $3 x + y \leq 15,4 x + 3 y \leq 30 , x \geq 0 , \text { and } y \geq 0$ .

Sketch the graph of the solution set of each system of inequalities. $2 x - 5 y < - 6$ $3 x + y < 8$

A total of $50,000 is invested in two funds paying 8% and 13% simple interest. The total yearly interest is $5100. How much is invested at the 8% rate?

Which of the following vertices of the constraint region shown is a maximum value of the objective function below. $z = - 2 x - y - 5$

Which of the following vertices of the constraint region shown is a minimum value of the objective function below. $z = - 6 x - 4 y + 2$

Use back-substitution to solve the system of linear equations. $\left\{ \begin{array} { r } - x + 7 y - 5 z = 51 \\5 y - 8 z = 82 \\z = - 9\end{array} \right.$

Sketch the graph of the solution set of each system of inequalities. $y > 2 x - 1$ $y > x ^ { 2 } - 2 x + 2$

Solve the system by the method of elimination. $\left\{ \begin{array} { l } 0.02 x - 0.02 y = 0.17 \\0.01 x + 0.05 y = 0.18\end{array} \right.$

Solve the system by the method of substitution. $\left\{ \begin{aligned}x ^ { 2 } + y ^ { 2 } & = 625 \\7 x - 24 y & = 0\end{aligned} \right.$

Solve the system of linear equations. $\left\{ \begin{aligned}- 3 x + 2 y - z & = 9 \\- x - 3 y + z & = 12 \\x - 2 y + 4 z & = 13\end{aligned} \right.$

Determine which ordered pair is a solution of the system. $\left\{ \begin{array} { l } 3 x - 4 y = 8 \\4 x - 7 y = 19\end{array} \right.$

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Equations and Inequalities

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Limits and Derivatives

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Exam 6: Systems of Equations and Inequalities

Sketch the graph of the inequality below. x<−3x < - 3x<−3

Find the minimum and maximum values of the objective function z=2x+8yz = 2 x + 8 yz=2x+8y and where they occur, subject to the constraints x≥0,y≥0,2x+y≤4x \geq 0 , y \geq 0,2 x + y \leq 4x≥0,y≥0,2x+y≤4 .

For the given supply and demand equations, find the consumer surplus. Round to the nearest dollar. Demand Supply P = 160 - 0.00006x p = 110 + 0.00003x

Maximize the object function z=4x+3yz = 4 x + 3 yz=4x+3y subject to the constraints 3x+y≤15,4x+3y≤30,x≥0, and y≥03 x + y \leq 15,4 x + 3 y \leq 30 , x \geq 0 , \text { and } y \geq 03x+y≤15,4x+3y≤30,x≥0, and y≥0 .

Sketch the graph of the solution set of each system of inequalities. 2x−5y<−62 x - 5 y < - 62x−5y<−6 3x+y<83 x + y < 83x+y<8

A total of $50,000 is invested in two funds paying 8% and 13% simple interest. The total yearly interest is $5100. How much is invested at the 8% rate?

Which of the following vertices of the constraint region shown is a maximum value of the objective function below. z=−2x−y−5z = - 2 x - y - 5z=−2x−y−5

Which of the following vertices of the constraint region shown is a minimum value of the objective function below. z=−6x−4y+2z = - 6 x - 4 y + 2z=−6x−4y+2

Use back-substitution to solve the system of linear equations. {−x+7y−5z=515y−8z=82z=−9\left\{ \begin{array} { r } - x + 7 y - 5 z = 51 \\5 y - 8 z = 82 \\z = - 9\end{array} \right.⎩⎨⎧​−x+7y−5z=515y−8z=82z=−9​

Sketch the graph of the solution set of each system of inequalities. y>2x−1y > 2 x - 1y>2x−1 y>x2−2x+2y > x ^ { 2 } - 2 x + 2y>x2−2x+2

Solve the system by the method of elimination. {0.02x−0.02y=0.170.01x+0.05y=0.18\left\{ \begin{array} { l } 0.02 x - 0.02 y = 0.17 \\0.01 x + 0.05 y = 0.18\end{array} \right.{0.02x−0.02y=0.170.01x+0.05y=0.18​

Solve the system by the method of substitution. {x2+y2=6257x−24y=0\left\{ \begin{aligned}x ^ { 2 } + y ^ { 2 } & = 625 \\7 x - 24 y & = 0\end{aligned} \right.{x2+y27x−24y​=625=0​

Solve the system of linear equations. {−3x+2y−z=9−x−3y+z=12x−2y+4z=13\left\{ \begin{aligned}- 3 x + 2 y - z & = 9 \\- x - 3 y + z & = 12 \\x - 2 y + 4 z & = 13\end{aligned} \right.⎩⎨⎧​−3x+2y−z−x−3y+zx−2y+4z​=9=12=13​

Determine which ordered pair is a solution of the system. {3x−4y=84x−7y=19\left\{ \begin{array} { l } 3 x - 4 y = 8 \\4 x - 7 y = 19\end{array} \right.{3x−4y=84x−7y=19​

Fundamental Concepts of Algebra

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Matrices and Determinants

Limits and Derivatives

Applications of the Derivative

Further Applications of the Derivative

Derivatives of Exponential and Logarithmic Functions

Integration and Its Applications

Techniques of Integration

Functions of Several Variables

Trigonometric Functions Web

Series and Taylor Polynomials Web

Probability Web

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Sketch the graph of the inequality below. $x < - 3$

Maximize the object function $z = 4 x + 3 y$ subject to the constraints $3 x + y \leq 15,4 x + 3 y \leq 30 , x \geq 0 , \text { and } y \geq 0$ .

Sketch the graph of the solution set of each system of inequalities. $2 x - 5 y < - 6$ $3 x + y < 8$

Which of the following vertices of the constraint region shown is a maximum value of the objective function below. $z = - 2 x - y - 5$

Which of the following vertices of the constraint region shown is a minimum value of the objective function below. $z = - 6 x - 4 y + 2$

Use back-substitution to solve the system of linear equations. $\left\{ \begin{array} { r } - x + 7 y - 5 z = 51 \\5 y - 8 z = 82 \\z = - 9\end{array} \right.$

Sketch the graph of the solution set of each system of inequalities. $y > 2 x - 1$ $y > x ^ { 2 } - 2 x + 2$

Solve the system by the method of elimination. $\left\{ \begin{array} { l } 0.02 x - 0.02 y = 0.17 \\0.01 x + 0.05 y = 0.18\end{array} \right.$

Solve the system by the method of substitution. $\left\{ \begin{aligned}x ^ { 2 } + y ^ { 2 } & = 625 \\7 x - 24 y & = 0\end{aligned} \right.$

Solve the system of linear equations. $\left\{ \begin{aligned}- 3 x + 2 y - z & = 9 \\- x - 3 y + z & = 12 \\x - 2 y + 4 z & = 13\end{aligned} \right.$

Determine which ordered pair is a solution of the system. $\left\{ \begin{array} { l } 3 x - 4 y = 8 \\4 x - 7 y = 19\end{array} \right.$