Question 1

Solve the problem.
-There is a relationship between the expected number of tickets sold for a raffle and the dollar value of the prize for the raffle. The equation T - 7P = 50 describes this relationship, where T is the expected number of
Tickets sold, and P is the dollar value of the raffle prize. Suppose the expected ticket sales for a certain
Raffle are 2150. Substitute 2150 into the equation to determine the dollar value of the raffle prize.
A)$300
B)$15,100
C)$250
D)$2100

Accepted Answer

The answer of Solve the problem.
-There is a relationship between...

Question 2

Solve the problem.
-A car rental agency charges $250 per week plus $0.10 per mile to rent a car. The total cost, C, for the renting the car for one week and driving it x miles can be modeled by the formula C = 0.10x + 250. How
Many miles can you travel in one week for $310?
A)600 miles
B)575 miles
C)281 miles
D)3100 miles

Accepted Answer

To find the number of miles you can travel for $310, substitute $310 for C in the equation and solve for x: $310 = 0.10x + 250. Subtract 250 from both sides to get 60 = 0.10x. Divide both sides by 0.10 to find x = 600 miles.

Question 3

Find all values of x satisfying the given conditions.
-$y _ { 1 } = \frac { 1 } { x + 5 } , y _ { 2 } = \frac { 1 } { x + 3 } , y _ { 3 } = \frac { - 2 } { x ^ { 2 } + 8 x + 15 }$, and the sum of $y _ { 1 }$ and 2 times $y _ { 2 }$ is $y 3$.
A) $\{ 0 \}$
B) $\{ - 5 \}$
C) $\{ 3 \}$
D) $\varnothing$

Accepted Answer

The given condition translates to $y_1 + 2y_2 = y_3$. Substituting the given expressions for $y_1$, $y_2$, and $y_3$ and simplifying does not yield a solution that satisfies the original equations, indicating that there are no values of $x$ that meet the given conditions.

Question 4

Solve the problem.
-The formula $\mathrm { y } = \frac { 28,000 + 220 \mathrm { x } } { \mathrm { x } }$ models the average cost per unit, $\mathrm { y }$, for Electrostuff to manufacture $x$ units of Electrogadget IV. How many units must the company produce to have an average cost per unit of $\$ 410$ ?
A) 147 units
B) 149 units
C) 116 units
D) 127 units

Accepted Answer

To have an average cost per unit of $410, the equation becomes:$$410 = \frac { 28,000 + 220 \mathrm { x } } { \mathrm { x } }$$$$410x = 28,000 + 220x$$$$190x = 28,000$$$$x = 147.37$$Since the company cannot produce a fraction of a unit, the company must produce 147 units to have an average cost per unit of $410.

Question 5

Use Linear Equations to Solve Problems
Use the five-step strategy for solving word problems to find the number or numbers described in the following
exercise.
-When 30% of a number is added to the number, the result is 260. What is the number?
A)200
B)60
C)23
D)130

Accepted Answer

Let the number be x. According to the problem, 30% of x added to x equals 260. This can be written as x + 0.30x = 260. Simplifying, we get 1.30x = 260. Solving for x, we find x = 260 / 1.30 = 200.

Question 6

Use Linear Equations to Solve Problems
Use the five-step strategy for solving word problems to find the number or numbers described in the following
exercise.
-When 5 times a number is subtracted from 7 times the number, the result is 22. What is the number?
A)11
B)2
C)-11
D)0.5

Accepted Answer

Let's call the unknown number "x". According to the problem, we have the equation 7x - 5x = 22. Simplifying this equation, we get 2x = 22, which means x = 11. Therefore, the answer is A) 11.

Question 7

Recognize Identities, Conditional Equations, and Inconsistent Equations
Determine whether the equation is an identity, a conditional equation, or an inconsistent equation.
-$$\frac { 7 } { y + 3 } - \frac { 2 } { y - 3 } = \frac { 3 } { y ^ { 2 } - 9 }$$
A)Identity
B)Conditional equation
C)Inconsistent equation

Accepted Answer

To determine whether the equation is an identity, a conditional equation, or an inconsistent equation, we can start by simplifying it.

First, we need to find a common denominator for the two fractions on the left side of the equation:

$$\frac { 7 } { y + 3 } - \frac { 2 } { y - 3 } = \frac { 3 } { y ^ { 2 } - 9 }$$

LCD: (y+3)(y-3)

$$\frac { 7(y-3) } { (y + 3)(y-3) } - \frac { 2(y+3) } { (y + 3)(y-3) } = \frac { 3 } { (y + 3)(y-3) }$$

Simplify:

$$\frac { 7y - 21 - 2y - 6 } { (y + 3)(y-3) } = \frac { 3 } { (y + 3)(y-3) }$$

$$\frac { 5y - 27 } { (y + 3)(y-3) } = \frac { 3 } { (y + 3)(y-3) }$$

Now, we can see that the denominators on both sides of the equation are the same, so we can compare the numerators. If the numerators are equal, then the equation is an identity. If the numerators are not equal but there is still some value of y that makes both sides of the equation equal, then the equation is a conditional equation. If there is no such value of y, then the equation is inconsistent.

In this case, we can see that the numerators are not equal, so it is not an identity. However, we can see that we can multiply both sides by the common denominator to eliminate the fractions:

$$5y - 27 = 3$$

$$5y = 30$$

$$y = 6$$

Since there is a specific value of y that makes both sides of the equation equal (y=6), the equation is a conditional equation. Therefore, the answer is choice B.

Question 8

Solve the problem.
-The U.S. Maritime Administration estimated that the cost per ton of building an oil tanker could be represented by the model $y = \frac { 101,000 } { x + 205 }$, where $y$ is the cost in dollars per ton and $x$ is the tons (in thousands). What size of oil tanker (in thousands of tons) can be built for $\$ 200$ per ton?
A) 300 thousand tons
B) 249 thousand tons
C) 710 thousand tons
D) 30 thousand tons

Accepted Answer

The answer of Solve the problem.
-The U.S. Maritime Administration estimated...

Question 9

Find all values of x satisfying the given conditions.
-$y _ { 1 } = x _ { 1 } y _ { 2 } = 9 + x _ { 1 } y _ { 3 } = 3 ( x - 7 ) + 10 x$, and the sum of 8 times $y _ { 1 }$ and 4 times $y _ { 2 }$ equals $y _ { 3 }$.
A) $\{ 57 \}$
B) $\{ 16 \}$
C) $\{ - 57 \}$
D) $\{ - 16 \}$

Accepted Answer

Explanation:$y _ { 1 } = x _ { 1 } y _ { 2 } = 9 + x _ { 1 } y _ { 3 } = 3 ( x - 7 ) + 10 x$$8y _ { 1 } + 4y _ { 2 } = y _ { 3 }$$8x _ { 1 } y _ { 2 } + 4y _ { 2 } = 9 + x _ { 1 } y _ { 3 }$$8x _ { 1 } y _ { 2 } + 4y _ { 2 } = 9 + x _ { 1 } ( 8x _ { 1 } y _ { 2 } + 4y _ { 2 } )$$8x _ { 1 } y _ { 2 } + 4y _ { 2 } = 9 + 8x _ { 1 } ^ { 2 } y _ { 2 } + 4x _ { 1 } y _ { 2 }$$8x _ { 1 } ^ { 2 } y _ { 2 } - 4x _ { 1 } y _ { 2 } - 4y _ { 2 } = - 9$$4y _ { 2 } ( 2x _ { 1 } ^ { 2 } - x _ { 1 } - 1 ) = - 9$$y _ { 2 } = \frac { 9 } { 4 ( 2x _ { 1 } ^ { 2 } - x _ { 1 } - 1 ) }$$y _ { 1 } = x _ { 1 } y _ { 2 } = \frac { 9x _ { 1 } } { 4 ( 2x _ { 1 } ^ { 2 } - x _ { 1 } - 1 ) }$$y _ { 3 } = 9 + x _ { 1 } y _ { 3 } = 9 + \frac { 9x _ { 1 } } { 4 ( 2x _ { 1 } ^ { 2 } - x _ { 1 } - 1 ) }$$8y _ { 1 } + 4y _ { 2 } = y _ { 3 }$$8 \cdot \frac { 9x _ { 1 } } { 4 ( 2x _ { 1 } ^ { 2 } - x _ { 1 } - 1 ) } + 4 \cdot \frac { 9 } { 4 ( 2x _ { 1 } ^ { 2 } - x _ { 1 } - 1 ) } = 9 + \frac { 9x _ { 1 } } { 4 ( 2x _ { 1 } ^ { 2 } - x _ { 1 } - 1 ) }$$18x _ { 1 } + 9 = 9 + 9x _ { 1 } + 4 ( 2x _ { 1 } ^ { 2 } - x _ { 1 } - 1 )$$18x _ { 1 } + 9 = 9 + 9x _ { 1 } + 8x _ { 1 } ^ { 2 } - 4x _ { 1 } - 4$$8x _ { 1 } ^ { 2 } - 5x _ { 1 } + 13 = 0$$( 2x _ { 1 } - 1 ) ( 4x _ { 1 } - 13 ) = 0$$x _ { 1 } = \frac { 1 } { 2 } , \frac { 13 } { 4 }$$y _ { 2 } = \frac { 9 } { 4 ( 2 \cdot \frac { 1 } { 2 } ^ { 2 } - \frac { 1 } { 2 } - 1 ) } = \frac { 9 } { 4 ( \frac { 1 } { 2 } - \frac { 1 } { 2 } - 1 ) } = \frac { 9 } { 4 ( - 1 ) } = - \frac { 9 } { 4 }$$y _ { 2 } = \frac { 9 } { 4 ( 2 \cdot \frac { 13 } { 4 } ^ { 2 } - \frac { 13 } { 4 } - 1 ) } = \frac { 9 } { 4 ( \frac { 169 } { 16 } - \frac { 13 } { 4 } - 1 ) } = \frac { 9 } { 4 ( \frac { 169 } { 16 } - \frac { 52 } { 16 } - \frac { 16 } { 16 } ) } = \frac { 9 } { 4 ( \frac { 101 } { 16 } ) } = \frac { 9 } { \frac { 404 } { 16 } } = \frac { 9 \cdot 16 } { 404 } = \frac { 144 } { 404 } = \frac { 36 } { 101 }$$y _ { 1 } = x _ { 1 } y _ { 2 } = \frac { 1 } { 2 } \cdot ( - \frac { 9 } { 4 } ) = - \frac { 9 } { 8 }$$y _ { 1 } = x _ { 1 } y _ { 2 } = \frac { 13 } { 4 } \cdot \frac { 36 } { 101 } = \frac { 468 } { 404 } = \frac { 117 } { 101 }$$y _ { 3 } = 9 + x _ { 1 } y _ { 3 } = 9 + \frac { 1 } { 2 } \cdot ( - \frac { 9 } { 4 } ) = 9 - \frac { 9 } { 8 } = \frac { 72 } { 8 } - \frac { 9 } { 8 } = \frac { 63 } { 8 }$$y _ { 3 } = 9 + x _ { 1 } y _ { 3 } = 9 + \frac { 13 } { 4 } \cdot \frac { 36 } { 101 } = 9 + \frac { 468 } { 404 } = 9 + \frac { 117 } { 101 } = \frac { 909 } { 101 } + \frac { 117 } { 101 } = \frac { 1026 } { 101 } = \frac { 342 } { 33 }$$8y _ { 1 } + 4y _ { 2 } = y _ { 3 }$$8 \cdot ( - \frac { 9 } { 8 } ) + 4 \cdot ( - \frac { 9 } { 4 } ) = \frac { 342 } { 33 }$$- 9 + ( - 9 ) = \frac { 342 } { 33 }$$- 18 = \frac { 342 } { 33 }$This is a contradiction, so there are no solutions.

Question 10

Solve the problem.
-The equation V = -2000t + 23,000 describes the value in dollars of a certain model of car after it is t years old. If a car is worth $15,000, substitute 15,000 into the equation to find the age of the car.
A)4 years
B)3 years
C)5 years
D)6 years

Accepted Answer

Substituting $15,000 for V in the equation gives 15000 = -2000t + 23000. Solving for t, we subtract 23000 from both sides to get -8000 = -2000t, then divide by -2000 to find t = 4.

Question 11

Use Linear Equations to Solve Problems
Use the five-step strategy for solving word problems to find the number or numbers described in the following
exercise.
-When four times the number is added to 7 times the number, the result is 33. What is the number?
A)3
B)-4.7
C)4.7
D)1

Accepted Answer

Let's use the five-step strategy to solve this problem:
Step 1: Understand the problem.
We need to find a number such that when we multiply it by 4 and add the result to 7 times the number, we get 33.

Step 2: Plan a solution.
Let's call the number we're looking for "x". Then we can write an equation based on the information given:
4x + 7x = 33
Simplifying this equation, we get:
11x = 33
Solving for x, we get:
x = 3

Step 3: Solve the problem.
We found that the number we're looking for is 3.

Step 4: Check the answer.
To check our answer, we can substitute x=3 into the original equation:
4(3) + 7(3) = 12 + 21 = 33
The left side of the equation matches the right side, so we can be confident that our answer is correct.

Step 5: Communicate the answer.
The number we're looking for is 3. Therefore, the answer is A.

Question 12

Find all values of x satisfying the given conditions.
-$\mathrm { y } _ { 1 } = \frac { 1 } { \mathrm { x } + 3 } , \mathrm { y } _ { 2 } = \frac { 1 } { \mathrm { x } - 3 } , \mathrm { y } _ { 3 } = \frac { 1 } { \mathrm { x } ^ { 2 } - 9 }$, and the difference between 8 times $\mathrm { y } _ { 1 }$ and 6 times $\mathrm { y } _ { 2 }$ is the product of 10 and $y _ { 3 }$.
A) $\{ 26 \}$
B) $\{ - 26 \}$
C) $\{ \sqrt { 47 } \}$
D) $\{ 52 \}$

Accepted Answer

The answer of Find all values of x satisfying the...

Question 13

Find all values of x satisfying the given conditions.
-$y _ { 1 } = 6 x , y _ { 2 } = ( 5 x - 1 )$, and $y _ { 1 }$ exceeds $y _ { 2 }$ by 2
A) $\{ 1 \}$
B) $\left\{ \frac { 1 } { 11 } \right\}$
C) $\{ - 1 \}$
D) $\left\{ - \frac { 1 } { 11 } \right\}$

Accepted Answer

The answer of Find all values of x satisfying the...

Question 14

Solve the problem.
-A local race for charity has taken place since 1993. Using the actual speeds of the winners from 1993 through 1998, mathematicians obtained the formula y = 0.18x + 5, in which x represents the number of
Years after 1993 and y represents the winning speed in miles per hour. In what year is the winning speed
Predicted to be 7.16 mph?
A)2005
B)2004
C)2006
D)2007

Accepted Answer

The winning speed is predicted to be 7.16 mph in 2005.To solve for x, substitute y = 7.16 into the equation y = 0.18x + 5 and solve for x:7.16 = 0.18x + 50.18x = 2.16x = 12Since x represents the number of years after 1993, the winning speed is predicted to be 7.16 mph in 1993 + 12 = 2005.

Question 15

Solve the problem.
-A certain store has a fax machine available for use by its customers. The store charges $2.35 to send the first page and $0.70 for each subsequent page. The total price, P, for the faxing x pages can be modeled by
The formula P = 0.70(x - 1)+ 2.35. Determine the number of pages that can be faxed for $9.35.
A)11 pages
B)57 pages
C)4 pages
D)13 pages

Accepted Answer

To find the number of pages that can be faxed for $9.35, we can use the given formula $P = 0.70(x - 1) + 2.35$. Setting $P = 9.35$, we get $9.35 = 0.70(x - 1) + 2.35$. Solving for $x$, we find that $x = 11$, meaning 11 pages can be faxed for $9.35.

Question 16

Use Linear Equations to Solve Problems
Use the five-step strategy for solving word problems to find the number or numbers described in the following
exercise.
-30% of what number is 96?
A)320
B)3200
C)32
D)28.8

Accepted Answer

Let the number be x. The equation is 0.30x = 96. Solving for x gives x = 96 / 0.30 = 320.

Question 17

Recognize Identities, Conditional Equations, and Inconsistent Equations
Determine whether the equation is an identity, a conditional equation, or an inconsistent equation.
-$$\frac { 1 } { x + 5 } + \frac { 3 } { x + 4 } = \frac { - 1 } { x ^ { 2 } + 9 x + 20 }$$
A)Identity
B)Conditional equation
C)Inconsistent equation

Accepted Answer

The equation is inconsistent because the left side simplifies to a common denominator of $(x+5)(x+4)$, which is the same as the denominator on the right side. However, the numerators do not match, leading to a contradiction.

Question 18

Use Linear Equations to Solve Problems
Use the five-step strategy for solving word problems to find the number or numbers described in the following
exercise.
-One number exceeds another by -12. The sum of the numbers is 2. What are the numbers?
A)-5 and 7
B)-6 and 8
C)5 and 8
D)No solution

Accepted Answer

Let the two numbers be x and y, with x being the larger number. According to the problem, x = y - 12 and x + y = 2. Solving these equations simultaneously, we find that the numbers are -5 and 7.

Question 19

Solve the problem.
-Suppose a cost-benefit model is given by $y = \frac { 2974 x } { 100 - x }$, where $y$ is the cost for removing $x$ percent of a given pollutant. What percent of pollutant can be removed for $\$ 30,000$ ? Round your answer to the nearest tenth of a percent.
A) $91.0 \%$
B) $111.0 \%$
C) $9.1 \%$
D) $502.2 \%$

Accepted Answer

To find the percent of pollutant that can be removed for $30,000, we need to set the given equation equal to $30,000 and solve for x:
\begin{align*}
30,000 &= \frac{2974x}{100-x}\
30,000(100-x) &= 2974x\
3,000,000-30,000x &= 2974x\
33074x &= 3,000,000\
x &= \frac{3,000,000}{33074} \approx 90.95\%
\end{align*}
Rounded to the nearest tenth, we get 91.0%. Therefore, the answer is A.

Question 20

Use Linear Equations to Solve Problems
Use the five-step strategy for solving word problems to find the number or numbers described in the following
exercise.
-When a number is decreased by 20% of itself, the result is 144. What is the number?
A)180
B)11
C)900
D)36

Accepted Answer

Let the number be x. Decreasing x by 20% of itself means subtracting 20% of x from x, which can be written as x - 0.20x = 144. Simplifying, 0.80x = 144. Solving for x gives x = 144 / 0.80 = 180.

Quiz 1: Equations and Inequalities

Solve the problem. -A car rental agency charges $250 per week plus $0.10 per mile to rent a car. The total cost, C, for the renting the car for one week and driving it x miles can be modeled by the formula C = 0.10x + 250. How Many miles can you travel in one week for $310?

Find all values of x satisfying the given conditions. - $y _ { 1 } = \frac { 1 } { x + 5 } , y _ { 2 } = \frac { 1 } { x + 3 } , y _ { 3 } = \frac { - 2 } { x ^ { 2 } + 8 x + 15 }$ , and the sum of $y _ { 1 }$ and 2 times $y _ { 2 }$ is $y 3$ .

Use Linear Equations to Solve Problems Use the five-step strategy for solving word problems to find the number or numbers described in the following exercise. -When 30% of a number is added to the number, the result is 260. What is the number?

Use Linear Equations to Solve Problems Use the five-step strategy for solving word problems to find the number or numbers described in the following exercise. -When 5 times a number is subtracted from 7 times the number, the result is 22. What is the number?

Recognize Identities, Conditional Equations, and Inconsistent Equations Determine whether the equation is an identity, a conditional equation, or an inconsistent equation. - $\frac { 7 } { y + 3 } - \frac { 2 } { y - 3 } = \frac { 3 } { y ^ { 2 } - 9 }$

Find all values of x satisfying the given conditions. - $y _ { 1 } = x _ { 1 } y _ { 2 } = 9 + x _ { 1 } y _ { 3 } = 3 ( x - 7 ) + 10 x$ , and the sum of 8 times $y _ { 1 }$ and 4 times $y _ { 2 }$ equals $y _ { 3 }$ .

Solve the problem. -The equation V = -2000t + 23,000 describes the value in dollars of a certain model of car after it is t years old. If a car is worth $15,000, substitute 15,000 into the equation to find the age of the car.

Use Linear Equations to Solve Problems Use the five-step strategy for solving word problems to find the number or numbers described in the following exercise. -When four times the number is added to 7 times the number, the result is 33. What is the number?

Find all values of x satisfying the given conditions. - $y _ { 1 } = 6 x , y _ { 2 } = ( 5 x - 1 )$ , and $y _ { 1 }$ exceeds $y _ { 2 }$ by 2

Use Linear Equations to Solve Problems Use the five-step strategy for solving word problems to find the number or numbers described in the following exercise. -30% of what number is 96?

Recognize Identities, Conditional Equations, and Inconsistent Equations Determine whether the equation is an identity, a conditional equation, or an inconsistent equation. - $\frac { 1 } { x + 5 } + \frac { 3 } { x + 4 } = \frac { - 1 } { x ^ { 2 } + 9 x + 20 }$

Use Linear Equations to Solve Problems Use the five-step strategy for solving word problems to find the number or numbers described in the following exercise. -One number exceeds another by -12. The sum of the numbers is 2. What are the numbers?

Solve the problem. -Suppose a cost-benefit model is given by $y = \frac { 2974 x } { 100 - x }$ , where $y$ is the cost for removing $x$ percent of a given pollutant. What percent of pollutant can be removed for $\$ 30,000$ ? Round your answer to the nearest tenth of a percent.

Use Linear Equations to Solve Problems Use the five-step strategy for solving word problems to find the number or numbers described in the following exercise. -When a number is decreased by 20% of itself, the result is 144. What is the number?