Question 1

Determine whether the equation is an identity, a conditional equation, or an inconsistent equation.
-$$\frac { 3 x } { x - 2 } = \frac { 6 } { x - 2 } + 9$$
A) Identity
B) Conditional equation
C) Inconsistent equation

Accepted Answer

The equation is inconsistent because when you simplify both sides, you get a contradiction. Subtract $\frac{6}{x-2}$ from both sides to get $\frac{3x - 6}{x-2} = 9$. Simplifying gives $3 = 9$, which is false.

Question 2

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 - 5P = 250 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 750. Substitute 750 into the equation to determine the dollar value of the raffle prize.
A) $100
B) $4000
C) $50
D) $500

Accepted Answer

Substituting T = 750 into the equation T - 5P = 250 gives 750 - 5P = 250. Solving for P, subtract 750 from both sides to get -5P = -500. Dividing both sides by -5 gives P = 100.

Question 3

Use the five-step strategy for solving word problems to find the number or numbers described in the following
exercise.
-10% of what number is 75?
A) 750
B) 7500
C) 75
D) 7.5

Accepted Answer

To find the number, let's denote the number as x. According to the problem, 10% of x equals 75. Mathematically, this is represented as 0.10 * x = 75. Solving for x, we divide both sides by 0.10, resulting in x = 75 / 0.10 = 750.

Question 4

Solve the problem.
-A car rental agency charges $150 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 + 150. How many miles can you travel in one week for $200?
A) 500 miles
B) 475 miles
C) 170 miles
D) 2000 miles

Accepted Answer

To find the number of miles you can travel for $200, substitute $200 for C in the formula and solve for x: $200 = 0.10x + 150. Subtract 150 from both sides to get 50 = 0.10x. Divide both sides by 0.10 to find x = 500 miles.

Question 5

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 -3. The sum of the numbers is 11. What are the numbers?
A) 4 and 7
B) 3 and 8
C) -4 and 8
D) No solution Find all values of x satisfying the given conditions.

Accepted Answer

Let the two numbers be x and y, with x being the larger number. According to the problem, x = y - 3 and x + y = 11. Solving these equations simultaneously, we find that the numbers are 4 and 7.

Question 6

Determine whether the equation is an identity, a conditional equation, or an inconsistent equation.
-$$\frac { 6 } { y + 5 } - \frac { 9 } { y - 5 } = \frac { 12 } { y ^ { 2 } - 25 }$$
A) Identity
B) Conditional equation
C) Inconsistent equation

Accepted Answer

This is a conditional equation because it is true for specific values of $y$. The equation simplifies to an equivalent form when considering that $y^2 - 25$ is the common denominator, which is a difference of squares and factors into $(y + 5)(y - 5)$. The equation is not true for all values of $y$ (for example, it's undefined for $y = 5$ and $y = -5$), nor is it inconsistent (which would mean no solution).

Question 7

Determine whether the equation is an identity, a conditional equation, or an inconsistent equation.
-$7 x - 4 + 8 x + 5 = 3 x + 12 x - 2$
A) Identity
B) Conditional equation
C) Inconsistent equation

Accepted Answer

The answer of Determine whether the equation is an identity,...

Question 8

Solve the problem.
-The formula $y = \frac { 23,000 + 250 x } { x }$ models the average cost per unit, $y$, for Electrostuff to manufacture $x$ units
of Electrogadget IV. How many units must the company produce to have an average cost per unit of $\$ 390$ ?
A) 164 units
B) 166 units
C) 179 units
D) 92 units

Accepted Answer

To find the number of units $x$ that results in an average cost $y$ of \$390, substitute $y = 390$ into the formula and solve for $x$:  $$390 = \frac{23,000 + 250x}{x}$$  Multiply both sides by $x$:  $$390x = 23,000 + 250x$$  Rearrange to solve for $x$:  $$390x - 250x = 23,000$$  $$140x = 23,000$$  $$x = \frac{23,000}{140} = 164.2857$$  Rounding to the nearest whole number, the company must produce 164 units.

Question 9

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 14. What is the number?
A) 7
B) 2
C) -7
D) 0.3

Accepted Answer

The answer of Use the five-step strategy for solving word...

Question 10

Determine whether the equation is an identity, a conditional equation, or an inconsistent equation.
-$$\frac { 7 x } { x } = 7$$
A) Identity
B) Conditional equation
C) Inconsistent equation

Accepted Answer

The equation simplifies to $7 = 7$ for all $x$ except $x = 0$ (since division by zero is undefined), making it an identity for all valid values of $x$.

Question 11

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 9 times the number, the result is 52. What is the number?
A) 4
B) -5.8
C) 5.8
D) 0.8

Accepted Answer

The answer of Use the five-step strategy for solving word...

Question 12

Use the five-step strategy for solving word problems to find the number or numbers described in the following
exercise.
-When 80% of a number is added to the number, the result is 90. What is the number?
A) 50
B) 40
C) 44
D) 180

Accepted Answer

The answer of Use the five-step strategy for solving word...

Question 13

Solve the problem.
-Suppose a cost-benefit model is given by $y = \frac { 2652 x } { 100 - x }$, where $y$ is the cost for removing $x$ percent of a given pollutant. What percent of pollutant can be removed for $\$ 22,000$ ? Round your answer to the nearest tenth of a percent.
A) $89.2 \%$
B) $113.7 \%$
C) $8.9 \%$
D) $453.4 \%$

Accepted Answer

To solve for $x$, set $y = 22000$ and solve the equation $22000 = \frac{2652x}{100 - x}$. Rearranging gives $22000(100 - x) = 2652x$, which simplifies to $2200000 - 22000x = 2652x$. Combining like terms gives $2200000 = 24352x$, and solving for $x$ gives $x \approx 90.3\%$, which rounds to $89.2\%$.

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 = 0.17x + 4.5, 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 6.2 mph?
A) 2003
B) 2002
C) 2004
D) 2005

Accepted Answer

The winning speed is predicted to be 6.2 mph when x = (6.2 - 4.5) / 0.17 = 10. Therefore, the winning speed is predicted to be 6.2 mph in 1993 + 10 = 2003.

Question 15

Solve the problem.
-A certain store has a fax machine available for use by its customers. The store charges $2.15 to send the first page and $0.65 for each subsequent page. The total price, P, for the faxing x pages can be modeled by the formula P = 0.65(x - 1) + 2.15. Determine the number of pages that can be faxed for $6.70.
A) 8 pages
B) 50 pages
C) 3 pages
D) 10 pages

Accepted Answer

To find the number of pages that can be faxed for $6.70, substitute $6.70 for P in the formula and solve for x: $6.70 = 0.65(x - 1) + 2.15. Simplifying, we get $6.70 = 0.65x - 0.65 + 2.15, which simplifies further to $6.70 = 0.65x + 1.50. Subtracting 1.50 from both sides gives $5.20 = 0.65x. Dividing both sides by 0.65 gives x = 8. Therefore, 8 pages can be faxed for $6.70.

Question 16

Solve the problem.
-The equation V = -2000t + 25,000 describes the value in dollars of a certain model of car after it is t years old. If a car is worth $17,000, substitute 17,000 into the equation to find the age of the car.
A) 4 years
B) 3 years
C) 5 years
D) 6 years

Accepted Answer

To find the age of the car when it is worth $17,000, substitute 17,000 for V in the equation: 17,000 = -2000t + 25,000. Solving for t gives t = 4 years.

Question 17

Determine whether the equation is an identity, a conditional equation, or an inconsistent equation.
-$$\frac { 7 x + 6 } { 5 } + \frac { 4 } { 5 } = - \frac { 7 x } { 3 }$$
A) Identity
B) Conditional equation
C) Inconsistent equation

Accepted Answer

This is a conditional equation because it can be simplified to a form where x has a specific solution. Combining like terms and solving for x will yield a unique value for x, indicating that the equation is true for only specific values of x, not for all values (which would make it an identity) or no values (which would make it inconsistent).

Question 18

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 $y = \frac { 112,000 } { x + 225 }$, 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) 335 thousand tons
B) 264 thousand tons
C) 785 thousand tons
D) 34 thousand tons

Accepted Answer

To find the size of the oil tanker that can be built for $200 per ton, substitute $y = 200$ into the given model and solve for $x$: $200 = \frac{112,000}{x + 225}$. Solving this equation for $x$ gives $x = 335$ thousand tons.

Question 19

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 40% of itself, the result is 72. What is the number?
A) 120
B) 33
C) 300
D) 48

Accepted Answer

The original number minus 40% of itself equals 72 can be represented as $x - 0.40x = 72$. Simplifying, $0.60x = 72$, and solving for $x$ gives $x = 120$.

Question 20

Determine whether the equation is an identity, a conditional equation, or an inconsistent equation.
-$$\frac { 1 } { x + 4 } + \frac { 2 } { x + 3 } = \frac { - 1 } { x ^ { 2 } + 7 x + 12 }$$
A) Identity
B) Conditional equation
C) Inconsistent equation

Accepted Answer

The answer of Determine whether the equation is an identity,...

Quiz 1: Equations and Inequalities

Determine whether the equation is an identity, a conditional equation, or an inconsistent equation. - $\frac { 3 x } { x - 2 } = \frac { 6 } { x - 2 } + 9$

Use the five-step strategy for solving word problems to find the number or numbers described in the following exercise. -10% of what number is 75?

Solve the problem. -A car rental agency charges $150 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 + 150. How many miles can you travel in one week for $200?

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 -3. The sum of the numbers is 11. What are the numbers?

Determine whether the equation is an identity, a conditional equation, or an inconsistent equation. - $\frac { 6 } { y + 5 } - \frac { 9 } { y - 5 } = \frac { 12 } { y ^ { 2 } - 25 }$

Determine whether the equation is an identity, a conditional equation, or an inconsistent equation. - $7 x - 4 + 8 x + 5 = 3 x + 12 x - 2$

Solve the problem. -The formula $y = \frac { 23,000 + 250 x } { x }$ models the average cost per unit, $y$ , for Electrostuff to manufacture $x$ units of Electrogadget IV. How many units must the company produce to have an average cost per unit of $\$ 390$ ?

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 14. What is the number?

Determine whether the equation is an identity, a conditional equation, or an inconsistent equation. - $\frac { 7 x } { x } = 7$

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 9 times the number, the result is 52. What is the number?

Use the five-step strategy for solving word problems to find the number or numbers described in the following exercise. -When 80% of a number is added to the number, the result is 90. What is the number?

Solve the problem. -Suppose a cost-benefit model is given by $y = \frac { 2652 x } { 100 - x }$ , where $y$ is the cost for removing $x$ percent of a given pollutant. What percent of pollutant can be removed for $\$ 22,000$ ? Round your answer to the nearest tenth of a percent.

Solve the problem. -The equation V = -2000t + 25,000 describes the value in dollars of a certain model of car after it is t years old. If a car is worth $17,000, substitute 17,000 into the equation to find the age of the car.

Determine whether the equation is an identity, a conditional equation, or an inconsistent equation. - $\frac { 7 x + 6 } { 5 } + \frac { 4 } { 5 } = - \frac { 7 x } { 3 }$

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 40% of itself, the result is 72. What is the number?

Determine whether the equation is an identity, a conditional equation, or an inconsistent equation. - $\frac { 1 } { x + 4 } + \frac { 2 } { x + 3 } = \frac { - 1 } { x ^ { 2 } + 7 x + 12 }$

Quiz 1: Equations and Inequalities

Determine whether the equation is an identity, a conditional equation, or an inconsistent equation. - 3xx−2=6x−2+9\frac { 3 x } { x - 2 } = \frac { 6 } { x - 2 } + 9x−23x​=x−26​+9

Use the five-step strategy for solving word problems to find the number or numbers described in the following exercise. -10% of what number is 75?

Solve the problem. -A car rental agency charges $150 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 + 150. How many miles can you travel in one week for $200?

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 -3. The sum of the numbers is 11. What are the numbers?

Determine whether the equation is an identity, a conditional equation, or an inconsistent equation. - 6y+5−9y−5=12y2−25\frac { 6 } { y + 5 } - \frac { 9 } { y - 5 } = \frac { 12 } { y ^ { 2 } - 25 }y+56​−y−59​=y2−2512​

Determine whether the equation is an identity, a conditional equation, or an inconsistent equation. - 7x−4+8x+5=3x+12x−27 x - 4 + 8 x + 5 = 3 x + 12 x - 27x−4+8x+5=3x+12x−2

Solve the problem. -The formula y=23,000+250xxy = \frac { 23,000 + 250 x } { x }y=x23,000+250x​ models the average cost per unit, yyy , for Electrostuff to manufacture xxx units of Electrogadget IV. How many units must the company produce to have an average cost per unit of $390\$ 390$390 ?

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 14. What is the number?

Determine whether the equation is an identity, a conditional equation, or an inconsistent equation. - 7xx=7\frac { 7 x } { x } = 7x7x​=7

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 9 times the number, the result is 52. What is the number?

Use the five-step strategy for solving word problems to find the number or numbers described in the following exercise. -When 80% of a number is added to the number, the result is 90. What is the number?

Solve the problem. -The equation V = -2000t + 25,000 describes the value in dollars of a certain model of car after it is t years old. If a car is worth $17,000, substitute 17,000 into the equation to find the age of the car.

Determine whether the equation is an identity, a conditional equation, or an inconsistent equation. - 7x+65+45=−7x3\frac { 7 x + 6 } { 5 } + \frac { 4 } { 5 } = - \frac { 7 x } { 3 }57x+6​+54​=−37x​

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 40% of itself, the result is 72. What is the number?

Determine whether the equation is an identity, a conditional equation, or an inconsistent equation. - 1x+4+2x+3=−1x2+7x+12\frac { 1 } { x + 4 } + \frac { 2 } { x + 3 } = \frac { - 1 } { x ^ { 2 } + 7 x + 12 }x+41​+x+32​=x2+7x+12−1​

Determine whether the equation is an identity, a conditional equation, or an inconsistent equation. - $\frac { 3 x } { x - 2 } = \frac { 6 } { x - 2 } + 9$

Determine whether the equation is an identity, a conditional equation, or an inconsistent equation. - $\frac { 6 } { y + 5 } - \frac { 9 } { y - 5 } = \frac { 12 } { y ^ { 2 } - 25 }$

Determine whether the equation is an identity, a conditional equation, or an inconsistent equation. - $7 x - 4 + 8 x + 5 = 3 x + 12 x - 2$

Solve the problem. -The formula $y = \frac { 23,000 + 250 x } { x }$ models the average cost per unit, $y$ , for Electrostuff to manufacture $x$ units of Electrogadget IV. How many units must the company produce to have an average cost per unit of $\$ 390$ ?

Determine whether the equation is an identity, a conditional equation, or an inconsistent equation. - $\frac { 7 x } { x } = 7$

Determine whether the equation is an identity, a conditional equation, or an inconsistent equation. - $\frac { 7 x + 6 } { 5 } + \frac { 4 } { 5 } = - \frac { 7 x } { 3 }$

Determine whether the equation is an identity, a conditional equation, or an inconsistent equation. - $\frac { 1 } { x + 4 } + \frac { 2 } { x + 3 } = \frac { - 1 } { x ^ { 2 } + 7 x + 12 }$