Deck 45: Systems of Inequalities

Question 1

Select the correct graph of the inequality.​ $$y > 7 x - 6$$ ​

A)​
B)​
C)​
D)​
E)​ ​

Answer 1

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Question 2

Select the correct graph of the inequality.​ $$y < 2 - x ^ { 2 }$$ ​

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B)​
C)​
D)​
E)​

Answer 2

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Question 3

Select the correct graph of the inequality.​ $$y \geq - 16.74 + 5.66 x$$ ​

A)​
B)​
C)​
D)​
E)​

Answer 3

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Question 4

Select the correct graph of the inequality.​ $$y < \ln 13 x$$ ​

A)​
B)​
C)​
D)​
E)​

Question 5

Select the correct graph of the inequality.​ $$3 y - x \geq 6$$ ​

A)​
B)​
C)​
D)​
E)​

Question 6

Select the correct graph of the inequality.​ $$6 \geq y$$ ​

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B)​
C)​
D)​
E)

Question 7

Select the correct graph of the inequality.​ $$y < - 1.8 x + 1.1$$ ​

A)
B)​ ​
C)​
D)​
E)​ ​

Question 8

Select the correct graph of the inequality.​ $$y < 4 ^ { - x - 5 }$$ ​

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B)​
C)​
D)​
E)​

Question 9

Select the correct graph of the inequality.​ $$y \leq \frac { 5 } { 1 + x ^ { 2 } }$$ ​

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B)​
C)​
D)​
E)​ ​

Question 10

Select the correct graph of the inequality.​ $$y < 4 - x$$ ​

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B)​
C)
D)​
E)​

Question 11

Select the correct graph of the inequality.​ $$y > - 4$$ ​

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B)​
C)​
D)​
E)​ ​

Question 12

Select the correct graph of the inequality.​ $$( x - 1 ) ^ { 2 } + ( y - 4 ) ^ { 2 } > 4$$ ​

A)​
B)​
C)​
D)​
E)​

Question 13

Select the correct graph of the inequality.​ $$( x + 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } < 9$$ ​

A)​
B)​
C)​
D)
E)​

Question 14

Select the correct graph of the inequality.​ $$x ^ { 2 } + 4 y - 8 \leq 0$$ ​

A)​
B)​
C)​
D)​
E)​

Question 15

Select the correct graph of the inequality.​ $$x \geq 3$$ ​

A)
B)​
C)​
D)​
E)​

Question 16

Select the correct graph of the inequality.​ $$4 y ^ { 2 } - x < 0$$ ​

A)​
B) ​
C)
D)​
E)​

Question 17

Select the correct graph of the inequality.​ $$4 x + 4 y \geq - 16$$ ​

A)​
B)​
C)​
D)​
E)​

Question 18

Select the correct graph of the inequality.​ $$y > \frac { - 21 } { x ^ { 2 } + x + 3 }$$ ​

A)​
B)​
C)​
D)​
E)​

Question 19

Select the correct graph of the inequality.​ $$y \geq - 4 - \ln ( x + 5 )$$ ​

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B)​
C)​
D)​
E)​

Question 20

Select the correct graph of the inequality.​ $$x < - 4$$ ​

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B)​
C)​
D)​
E)​

Question 21

For a concert event,there are $30 reserved seat tickets and $20 general admission tickets.There are 2000 reserved seats available,and fire regulations limit the number of paid ticket holders to 3000.The promoter must take in at least $65,000 in ticket sales.Find and graph a system of inequalities describing the different numbers of tickets that can be sold. ​

A) $$\left\{ \begin{array} { c } x + y \leq 3000 \\30 x + 20 y \geq 72,000 \\x \leq 2000 \\x \geq 0 \\y \geq 0\end{array} \right.$$ ​
B) $$\left\{ \begin{array} { c } x + y \leq 3000 \\30 x + 20 y \geq 68,000 \\x \leq 2000 \\x \geq 0 \\y \geq 0\end{array} \right.$$ ​
C)​ $$\left\{ \begin{array} { c } x + y \leq 3000 \\30 x + 20 y \geq 66,000 \\x \leq 2000 \\x \geq 0 \\y \geq 0\end{array} \right.$$
D) $$\left\{ \begin{array} { c } x + y \leq 3000 \\30 x + 20 y \geq 73,000 \\x \leq 2000 \\x \geq 0 \\y \geq 0\end{array} \right.$$ ​
E) $$\left\{ \begin{array} { c } x + y \leq 3000 \\30 x + 20 y \geq 65,000 \\x \leq 2000 \\x \geq 0 \\y \geq 0\end{array} \right.$$ ​

Question 22

Find the consumer surplus and producer surplus. ​
Demand $p = 600 - 0.0002 x$ ​
Supply $p = 450 + 0.0001 x$ ​

A)Consumer surplus: $25,000,150 Producer surplus: $12,500,400
B)Consumer surplus: $25,000,200 Producer surplus: $12,500,300
C)Consumer surplus: $25,000,100 Producer surplus: $12,500,100
D)Consumer surplus: $25,000,050 Producer surplus: $12,500,200
E)Consumer surplus: $25,000,000 Producer surplus: $12,500,000

Question 23

A dietitian is asked to design a special dietary supplement using two different foods.Each ounce of food X contains 20 units of calcium,15 units of iron,and 10 units of vitamin B.Each ounce of food Y contains 10 units of calcium,10 units of iron,and 20 units of vitamin B.The minimum daily requirements of the diet are 350 units of calcium,300 units of iron,and 400 units of vitamin B.Sketch a graph of the region corresponding to the system inequalities describing the different amounts of food X and food Y that can be used.​ $$\left\{ \begin{array} { c } 20 x + 10 y \geq 350 \\15 x + 10 y \geq 300 \\10 x + 20 y \geq 400 \\x \geq 0 \\y \geq 0\end{array} \right.$$ ​

A)​
B)​
C)
D)​
E)​

Question 24

Select the correct graph of the inequality.​ $$10 x ^ { 2 } - y - 11 > 0$$ ​

A)​
B)​
C)​
D)​
E)​

Question 25

A person plans to invest up to $26,000 in two different interest-bearing accounts.Each account is to contain at least $5000.Moreover,the amount in one account should be at least twice the amount in the other account.Find and graph a system of inequalities to describe the various amounts that can be deposited in each account. ​

A)​ $$\left\{ \begin{array} { c } x + y = 25,000 \\y \geq 2 x \\x \geq 5000 \\y \geq 5000\end{array} \right.$$
B) $$\left\{ \begin{array} { c } x + y \leq 26,000 \\y \geq 2 x \\x \geq 5000 \\y \geq 5000\end{array} \right.$$ ​
C) $$\left\{ \begin{array} { c } x + y = 22,000 \\y \geq 2 x \\x \geq 5000 \\y \geq 5000\end{array} \right.$$ ​
D) $$\left\{ \begin{array} { c } x + y \geq 30,000 \\y \geq 2 x \\x \geq 5000 \\y \geq 5000\end{array} \right.$$ ​
E) $$\left\{ \begin{array} { c } x + y \geq 23,000 \\y \geq 2 x \\x \geq 5000 \\y \geq 5000\end{array} \right.$$ ​

Question 26

Select an inequality for the shaded region shown in the figure.​ ​

A) $y \geq x ^ { 2 } - 3$
B) $y \geq x ^ { 2 } - 4$
C) $y \geq x ^ { 2 } - 2$
D) $y \geq x ^ { 2 } - 5$
E) $y \geq x ^ { 2 } - 6$

Question 27

A store sells two models of laptop computers.Because of the demand,the store stocks at least twice as many units of model A as of model B.The costs to the store for the two models are $800 and $1200,respectively.The management does not want more than $29,000 in computer inventory at any one time,and it wants at least four model A laptop computers and two model B laptop computers in inventory at all times.Find and graph a system of inequalities describing all possible inventory levels.
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A) $$\left\{\begin{array} { c } x \geq 2 y \\8 x + 12 y \leq 310 \\x \geq 4 \\y \geq 2\end{array}\right.$$
B) $$\left\{\begin{array} { c } x \geq 2 y \\8 x + 12 y \leq 290 \\x \geq 4 \\y \geq 2\end{array}\right.$$ ​
C) $$\left\{ \begin{array} { c } x \geq y \\8 x + 12 y \leq 290 \\x \geq 4 \\y \geq 2\end{array} \right.$$ ​
D)​ $$\left\{\begin{array} { c } x \geq 2 y \\8 x + 12 y \leq 290 \\x \geq 2 \\y \geq 2\end{array}\right.$$
E) $$\left\{ \begin{array} { c } x \geq 3 y \\8 x + 12 y \leq 290 \\x \geq 4 \\y \geq 2\end{array} \right.$$ ​

Question 28

Find the consumer surplus and producer surplus. ​
Demand $p = 70 - 0.5 x$ ​
Supply $p = 0.125 x$ ​

A)Consumer surplus: $3136 Producer surplus: $784
B)Consumer surplus: $3286 Producer surplus: $784
C)Consumer surplus: $3236 Producer surplus: $784
D)Consumer surplus: $3186 Producer surplus: $784
E)Consumer surplus: $3336 Producer surplus: $784

Question 29

A small company that manufactures two models of exercise machines has an order for 17 units of the standard model and 18 units of the deluxe model.The company has trucks of two different sizes that can haul the products,as shown in the table. $$\begin{array} { | l | l | l | } \hline \text { Truck } & \text { Standard } & \text { Deluxe } \\\hline \text { Large } & 6 & 3 \\\hline \text { Medium } & 4 & 6 \\\hline\end{array}$$
Find and graph a system of inequalities describing the numbers of trucks of each size that are needed to deliver the order.

A) $$\left\{ \begin{array} { c } 6 x + 4 y \geq 17 \\3 x + 6 y \geq 18 \\x \geq 0 \\y \geq 0\end{array} \right.$$ ​
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B) $$\left\{ \begin{array} { c } 6 x + 4 y \geq 16 \\3 x + 6 y \geq 18 \\x \geq 0 \\y \geq 0\end{array} \right.$$ ​
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C) $$\left\{ \begin{array} { c } 6 x + 4 y \geq 18 \\3 x + 6 y \geq 18 \\x \geq 0 \\y \geq 0\end{array} \right.$$ ​
D) $$\left\{ \begin{array} { c } 6 x + 4 y \geq 14 \\3 x + 6 y \geq 18 \\x \geq 0 \\y \geq 0\end{array} \right.$$ ​
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E)​ $$\left\{ \begin{array} { c } 6 x + 4 y \geq 20 \\3 x + 6 y \geq 18 \\x \geq 0 \\y \geq 0\end{array} \right.$$ ​

Question 30

Write a set of inequalities to describe the region. ​
Rectangle: vertices at $$( 1,5 ) , ( 9,5 ) , ( 9,9 ) , ( 1,9 )$$ ​

A) $$\left\{ \begin{array} { l } x \geq 1 \\x \leq 9 \\y \geq 5 \\y \leq 9\end{array} \right.$$
B) $$\left\{ \begin{array} { l } x \geq 4 \\x \leq 5 \\y \geq 9 \\y \leq 5\end{array} \right.$$
C) $$\left\{ \begin{array} { l } x \geq 3 \\x \leq 5 \\y \geq 9 \\y \leq 5\end{array} \right.$$
D) $$\left\{ \begin{array} { l } x \geq 2 \\x \leq 5 \\y \geq 9 \\y \leq 5\end{array} \right.$$
E) $$\left\{ \begin{array} { l } x \geq 5 \\x \leq 5 \\y \geq 9 \\y \leq 9\end{array} \right.$$

Question 31

Use a graphing utility to graph the inequality.​ $$y \geq \frac { 5 } { 9 } x - 4$$ ​

A) ​
B)​
C)
D) ​
E)​ ​

Question 32

A warehouse supervisor is told to ship at least 50 packages of gravel that weigh 55 pounds each and at least 40 bags of stone that weigh 70 pounds each.The maximum weight capacity of the truck to be used is 9,500 pounds.Find and graph a system of inequalities describing the numbers of bags of stone and gravel that can be shipped.
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A) $$\left\{ \begin{array} { c } 55 x + 70 y \leq 9,500 \\x \geq 50 \\y \geq 40\end{array} \right.$$ ​
B)​ $$\left\{ \begin{array} { c } 55 x + 70 y \leq 7,000 \\x \geq 50 \\y \geq 40\end{array} \right.$$
C) $$\left\{ \begin{array} { c } 55 x + 70 y \leq 8,000 \\x \geq 50 \\y \geq 40\end{array} \right.$$ ​
D)​ $$\left\{ \begin{array} { c } 55 x + 70 y \leq 10,000 \\x \geq 50 \\y \geq 40\end{array} \right.$$
E) $$\left\{ \begin{array} { c } 55 x + 70 y \leq 8,500 \\x \geq 50 \\y \geq 40\end{array} \right.$$ ​
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Question 33

Select a set of inequalities to describe the region.​ ​

A) $$\left\{ \begin{array} { c } y < 3 - 2 x \\y \geq x - 4 \\x \geq 1\end{array} \right.$$
B) $$\left\{ \begin{array} { c } y < 6 - 2 x \\y \geq x - 2 \\x \geq 1\end{array} \right.$$
C) $$\left\{ \begin{array} { c } y < 4 - 2 x \\y \geq x - 5 \\x \geq 1\end{array} \right.$$
D) $$\left\{ \begin{array} { c } y < 7 - 2 x \\y \geq x - 3 \\x \geq 1\end{array} \right.$$
E) $$\left\{ \begin{array} { c } y < 5 - 2 x \\y \geq x - 6 \\x \geq 1\end{array} \right.$$

Question 34

Select an inequality for the shaded region shown in the figure.​ ​

A) $x ^ { 2 } + y ^ { 2 } < 25$
B) $x ^ { 2 } + y ^ { 2 } < 4$
C) $x ^ { 2 } + y ^ { 2 } < 9$
D) $x ^ { 2 } + y ^ { 2 } < 1$
E) $x ^ { 2 } + y ^ { 2 } < 16$

Question 35

Select an inequality to describe the region.​ ​

A) $x ^ { 2 } + y ^ { 2 } \geq 4$
B) $x ^ { 2 } + y ^ { 2 } \geq 9$
C) $x ^ { 2 } + y ^ { 2 } \geq 1$
D) $x ^ { 2 } + y ^ { 2 } \geq 16$
E) $x ^ { 2 } + y ^ { 2 } \geq 25$

Question 36

Find the consumer surplus and producer surplus. ​
Demand $p = 100 - 0.03 x$ ​
Supply $p = 35 + 0.1 x$ ​

A)Consumer surplus: $3,750 Producer surplus: $12,800
B)Consumer surplus: $3,750 Producer surplus: $12,500
C)Consumer surplus: $3,750 Producer surplus: $12,600
D)Consumer surplus: $3,750 Producer surplus: $12,900
E)Consumer surplus: $3,750 Producer surplus: $12,700

Question 37

Select a set of inequalities to describe the region.​ ​

A) $$\left\{ \begin{array} { c } x \geq 0 \\y \geq 0 \\x ^ { 2 } + y ^ { 2 } < 9\end{array} \right.$$
B) $$\left\{ \begin{array} { c } x \geq 0 \\y \geq 0 \\x ^ { 2 } + y ^ { 2 } < 1\end{array} \right.$$
C) $$\left\{ \begin{array} { c } x \geq 0 \\y \geq 0 \\x ^ { 2 } + y ^ { 2 } < 4\end{array} \right.$$
D) $$\left\{ \begin{array} { c } x \geq 0 \\y \geq 0 \\x ^ { 2 } + y ^ { 2 } < 25\end{array} \right.$$
E) $$\left\{ \begin{array} { c } x \geq 0 \\y \geq 0 \\x ^ { 2 } + y ^ { 2 } < 16\end{array} \right.$$

Question 38

A dietitian is asked to design a special dietary supplement using two different foods.Each ounce of food X contains 20 units of calcium,15 units of iron,and 10 units of vitamin B.Each ounce of food Y contains 10 units of calcium,10 units of iron,and 20 units of vitamin B.The minimum daily requirements of the diet are 350 units of calcium,300 units of iron,and 400 units of vitamin B.Write a system of inequalities describing the different amounts of food X and food Y that can be used. ​

A) $$\left\{ \begin{array} { c } 20 x + 10 y \geq 350 \\15 x + 10 y \geq 300 \\10 x + 20 y \geq 400 \\x \geq 0 \\y \geq 0\end{array} \right.$$
B) $$\left\{ \begin{array} { c } 20 x + 10 y \geq 400 \\15 x + 10 y \geq 350 \\10 x + 20 y \geq 300 \\x \geq 0 \\y \geq 0\end{array} \right.$$
C) $$\left\{ \begin{array} { c } 20 x + 10 y \geq 300 \\15 x + 10 y \geq 350 \\10 x + 20 y \geq 400 \\x \geq 0 \\y \geq 0\end{array} \right.$$
D) $$\left\{ \begin{array} { c } 20 x + 10 y \geq 400 \\15 x + 10 y \geq 300 \\10 x + 20 y \geq 350 \\x \geq 0 \\y \geq 0\end{array} \right.$$
E) $$\left\{ \begin{array} { c } 20 x + 10 y \geq 300 \\15 x + 10 y \geq 400 \\10 x + 20 y \geq 350 \\x \geq 0 \\y \geq 0\end{array} \right.$$

Question 39

Select an inequality for the shaded region shown in the figure.​ ​

A) $y < 6 x + 7$
B) $y < 4 x + 5$
C) $y < 5 x + 6$
D) $y < 2 x + 3$
E) $y < 3 x + 4$

Question 40

Find the consumer surplus and producer surplus. ​
Demand $p = 100 - 0.00006 x$ ​
Supply $p = 90 + 0.00004 x$ ​

A)Consumer surplus: $300,100 Producer surplus: $ 200,100
B)Consumer surplus: $300,050 Producer surplus: $200,200
C)Consumer surplus: $300,200 Producer surplus: $200,300
D)Consumer surplus: $300,150 Producer surplus: $200,400
E)Consumer surplus: $300,000 Producer surplus: $200,000

Question 41

Use a graphing utility to graph the inequalities.Shade the region representing the solution set of the system. ​​ $$\left\{ \begin{array} { l } y > x ^ { 2 } - 5 x + 2 \\y < - x ^ { 2 } + x + 2\end{array} \right.$$ ​

A)​ 10 -10​
B)10 -10​
C)​10 ​ -10
D)​ 10 -10​
E)10​ ​ -10

Question 42

Sketch the graph and label the vertices of the solution set of the system of inequalities.Shade the solution set. $$\left\{ \begin{array} { l } x ^ { 2 } + y ^ { 2 } < 16 \\x ^ { 2 } + y ^ { 2 } \geq 25\end{array} \right.$$

A) ​
B)​ ​
C) $$\text { no solution }$$
D)​ ​
E)​

Question 43

Determine which one of the ordered pairs below is a solution of the system of linear inequalities.​ $$\left\{ \begin{array} { c } 2 x + 3 y > 2 \\- y - \frac { 1 } { 3 } x \leq - 3 \\13 x - 6 y > 0\end{array} \right.$$ ​

A)(2,8)
B)(6,1)
C)(0,0)
D)(-1,4)
E)(3,-3)

Question 44

For the given supply and demand equations,find the consumer surplus. Demand
Supply $p = 100 - 0.00006 x$ $p = 80 + 0.00002 x$ ​
​

A)$1,875,000
B)$2,812,500
C)$2,062,500
D)$2,250,000
E)$625,000

Question 45

Write an inequality for the shaded region shown in the figure.

A) $y \geq$ $( x - 2 ) ^ { 2 }$
B)y ≥ $x ^ { 2 } - 2$
C)y ≤ $2 - x ^ { 2 }$
D)y ≥ $2 - x ^ { 2 }$
E)y ≥ $x ^ { 2 } + 2$

Question 46

Use a graphing utility to graph the inequality.Shade the region representing the solution.​ $$y$$ ≤ $$2 ^ { 1 - x }$$

A)​10 -10
B)​ 10 -10
C)10​ -10
D)10 -10
E)​ 10 -10

Question 47

Determine which one of the ordered pairs below is a solution of the system of linear inequalities.​ $$\left\{ \begin{array} { c } x > - 3 \\y > - 1 \\y \leq 4 - \frac { 2 x } { 3 }\end{array} \right.$$ ​

A)(-2,2)
B)(3,-3)
C)(-5,-9)
D)(4,-7)
E)(9,8)

Question 48

Derive a set of inequalities to describe the region.​

A)​ $$\left\{ \begin{array} { c } x ^ { 2 } + y ^ { 2 } \leq 9 \\x ^ { 2 } + y ^ { 2 } \geq 16 \\x \geq 0 \\y \geq 0\end{array} \right.$$
B)​ $$\left\{ \begin{array} { c } x ^ { 2 } + y ^ { 2 } \geq 3 \\x ^ { 2 } + y ^ { 2 } \leq 4 \\x \geq 0 \\y \geq 0\end{array} \right.$$
C)​ $$\left\{ \begin{array} { l } x ^ { 2 } + y ^ { 2 } \geq 3 \\x ^ { 2 } + y ^ { 2 } \leq 4\end{array} \right.$$
D)​ $$\left\{ \begin{array} { c } x ^ { 2 } + y ^ { 2 } \geq 9 \\x ^ { 2 } + y ^ { 2 } \leq 16\end{array} \right.$$
E) $$\left\{ \begin{array} { c } x ^ { 2 } + y ^ { 2 } \geq 9 \\x ^ { 2 } + y ^ { 2 } \leq 16 \\x \geq 0 \\y \geq 0\end{array} \right.$$

Question 49

​Derive a set of inequalities to describe the region. ​
Triangle: vertices at (0,0), (4,0), (4,5)
​

A)​ $$\left\{ \begin{array} { c } y \geq \frac { 5 } { 4 } x \\y \geq 0 \\x \leq 4\end{array} \right.$$
B)​ $$\left\{ \begin{array} { c } y \leq \frac { 5 } { 4 } x \\x \geq 4\end{array} \right.$$
C)​ $$\left\{ \begin{array} { c } y \geq \frac { 4 } { 5 } x \\y \geq 0 \\x \leq 4\end{array} \right.$$
D)​ $$\left\{ \begin{array} { c } y \leq \frac { 5 } { 4 } x \\y \geq 0 \\x \leq 4\end{array} \right.$$
E)​ $$\left\{ \begin{array} { c } y \leq \frac { 5 } { 4 } x \\x \leq 4\end{array} \right.$$

Question 50

Sketch the graph of the inequality. $$( x - 3 ) ^ { 2 } + ( y + 1 ) ^ { 2 }$$ < 4

A)​ ​
B) ​
C) ​
D) ​
E) ​ ​