Question 1

Write the sentence as an equation. Let the variable x represent the number.
-The product of 6 and a number, increased by 3, is 21.

Accepted Answer

The sentence describes an operation where 6 is multiplied by a number (6x), and then 3 is added to the result (6x + 3). This sum equals 21, which matches option A.

Question 2

Write the sentence as an equation. Let the variable x represent the number.
-Three times a number is equal to 20 decreased by the number.

Accepted Answer

The correct equation is 3x = 20 - x, which represents "three times a number is equal to 20 decreased by the number."

Question 3

Solve the equation.
-x(5x - 1)= (5x + 1)(x - 3)

Accepted Answer

The answer of Solve the equation.
-x(5x - 1)= (5x +...

Question 4

Solve the equation.
-$$x \left( x ^ { 2 } + 8 \right) = 3 + x ^ { 3 }$$

Accepted Answer

Expanding the left side of the equation gives $x^3 + 8x = 3 + x^3$. Subtracting $x^3$ from both sides simplifies to $8x = 3$, leading to $x = \frac{3}{8}$.

Question 5

Solve the equation.
-$$\frac { 6 } { x + 4 } - \frac { 8 } { x - 4 } = \frac { 8 } { x ^ { 2 } - 16 }$$

Accepted Answer

The answer of Solve the equation.
-\[\frac { 6 } {...

Question 6

Solve the equation.
-0.50x - 0.20(20 + x)= 1.00(20)

Accepted Answer

The answer of Solve the equation.
-0.50x - 0.20(20 + x)=...

Question 7

Solve the equation.
-$$\frac { x } { x - 4 } - 4 = \frac { 4 } { x - 4 }$$

Accepted Answer

The equation simplifies to a contradiction, indicating no solution exists.

Question 8

Solve the problem.
-A rectangular carpet has a perimeter of 288 inches. The length of the carpet is 96 inches more than the width. What are the dimensions of the carpet?

Accepted Answer

The answer of Solve the problem.
-A rectangular carpet has a...

Question 9

Solve the equation.
-$$\frac { - 3 } { x - 7 } = \frac { 6 } { x - 14 } - \frac { - 21 } { x ^ { 2 } - 21 x + 98 }$$

Accepted Answer

The equation involves fractions with variables in the denominators. The denominators $x-7$, $x-14$, and $x^2-21x+98$ suggest potential restrictions on $x$. Specifically, $x^2-21x+98$ can be factored into $(x-7)(x-14)$, indicating that both 7 and 14 make at least one denominator zero, leading to division by zero, which is undefined. Therefore, there are no values of $x$ that satisfy the equation without causing division by zero.

Question 10

Write the sentence as an equation. Let the variable x represent the number.
-Three less than 4 times a number equals 17.

Accepted Answer

The correct equation is "4x - 3 = 17" because it translates to "three less than 4 times a number equals 17."

Question 11

Solve the problem.
-A bank loaned out $69,000, part of it at the rate of 15% per year and the rest at a rate of 8% per year. If the interest received was $7480, how much was loaned at 15%?

Accepted Answer

The answer of Solve the problem.
-A bank loaned out $69,000,...

Question 12

Solve the equation.
-$$\frac { 2 x } { x ^ { 2 } - 1 } = \frac { 3 } { x ^ { 2 } - 1 } - \frac { 4 } { x + 1 }$$

Accepted Answer

The answer of Solve the equation.
-\[\frac { 2 x }...

Question 13

Solve the equation.
-$$( x + 2 ) ( x - 1 ) = ( x + 1 ) ^ { 2 }$$

Accepted Answer

The answer of Solve the equation.
-\[( x + 2 )...

Question 14

Solve the problem.
-Center City East Parking Garage has a capacity of 253 cars more than Center City West Parking Garage. If the combined capacity for the two garages is 1219 cars, find the capacity for each garage.

Accepted Answer

Let the capacity of Center City West be x. Then, Center City East is x + 253. The equation is x + (x + 253) = 1219. Solving gives x = 483, so West is 483 cars and East is 736 cars.

Question 15

Solve the problem.
-Mary and her brother John collect foreign coins. Mary has four times the number of coins that John has. Together they have 150 foreign coins. Find how many coins Mary has.

Accepted Answer

Mary has 120 coins. If Mary has four times the number of coins John has, let's denote John's coins as x. Therefore, Mary's coins are 4x. Together, they have x + 4x = 5x = 150 coins. Solving for x gives x = 30 (John's coins), so Mary has 4x = 4*30 = 120 coins.

Question 16

Solve the problem.
-During an intramural basketball game, Team A scored 19 fewer points than Team B. Together, both teams scored a total of 151 points. How many points did Team A score during the game?

Accepted Answer

Team A scored 66 points. If Team B scored 19 more points than Team A, and together they scored 151 points, you can set up the equation: A + B = 151, where B = A + 19. Substituting B in the equation gives A + (A + 19) = 151, which simplifies to 2A + 19 = 151. Solving for A gives 2A = 132, so A = 66.

Question 17

Solve the problem.
-Don James wants to invest $59,000 to earn $6170 per year. He can invest in B-rated bonds paying 13% per year or in a Certificate of Deposit (CD)paying 7% per year. How much money should be invested in each to realize exactly $6170 in interest per year?

Accepted Answer

To earn $6170, let x be the amount in B-rated bonds at 13%, and y be the amount in CD at 7%. The equations are: x + y = $59,000 and 0.13x + 0.07y = $6170. Solving these gives x = $34,000 and y = $25,000.

Question 18

Write the sentence as an equation. Let the variable x represent the number.
-The product of 4 and a number increased by 3 is 20.

Accepted Answer

The correct equation that represents the sentence is "4(x + 3) = 20," which means "the product of 4 and a number increased by 3 equals 20."

Question 19

Write the sentence as an equation. Let the variable x represent the number.
-The sum of twice a number and 5 is 9.

Accepted Answer

The sentence "The sum of twice a number and 5 is 9" translates to the equation 2x + 5 = 9, where x is the variable representing the number.

Question 20

Solve the equation.
-$$\frac { 6 } { 5 x } - \frac { 1 } { x + 1 } = \frac { 3 } { x ( 2 x + 2 ) }$$

Accepted Answer

The answer of Solve the equation.
-\[\frac { 6 } {...

Quiz 1: Equations, Inequalities, and Applications

Write the sentence as an equation. Let the variable x represent the number. -The product of 6 and a number, increased by 3, is 21.

Write the sentence as an equation. Let the variable x represent the number. -Three times a number is equal to 20 decreased by the number.

Solve the equation. -x(5x - 1) = (5x + 1) (x - 3)

Solve the equation. - $x \left( x ^ { 2 } + 8 \right) = 3 + x ^ { 3 }$

Solve the equation. - $\frac { 6 } { x + 4 } - \frac { 8 } { x - 4 } = \frac { 8 } { x ^ { 2 } - 16 }$

Solve the equation. -0.50x - 0.20(20 + x) = 1.00(20)

Solve the equation. - $\frac { x } { x - 4 } - 4 = \frac { 4 } { x - 4 }$

Solve the problem. -A rectangular carpet has a perimeter of 288 inches. The length of the carpet is 96 inches more than the width. What are the dimensions of the carpet?

Solve the equation. - $\frac { - 3 } { x - 7 } = \frac { 6 } { x - 14 } - \frac { - 21 } { x ^ { 2 } - 21 x + 98 }$

Write the sentence as an equation. Let the variable x represent the number. -Three less than 4 times a number equals 17.

Solve the problem. -A bank loaned out $69,000, part of it at the rate of 15% per year and the rest at a rate of 8% per year. If the interest received was $7480, how much was loaned at 15%?

Solve the equation. - $\frac { 2 x } { x ^ { 2 } - 1 } = \frac { 3 } { x ^ { 2 } - 1 } - \frac { 4 } { x + 1 }$

Solve the equation. - $( x + 2 ) ( x - 1 ) = ( x + 1 ) ^ { 2 }$

Solve the problem. -Center City East Parking Garage has a capacity of 253 cars more than Center City West Parking Garage. If the combined capacity for the two garages is 1219 cars, find the capacity for each garage.

Solve the problem. -Mary and her brother John collect foreign coins. Mary has four times the number of coins that John has. Together they have 150 foreign coins. Find how many coins Mary has.

Solve the problem. -During an intramural basketball game, Team A scored 19 fewer points than Team B. Together, both teams scored a total of 151 points. How many points did Team A score during the game?

Solve the problem. -Don James wants to invest $59,000 to earn $6170 per year. He can invest in B-rated bonds paying 13% per year or in a Certificate of Deposit (CD) paying 7% per year. How much money should be invested in each to realize exactly $6170 in interest per year?

Write the sentence as an equation. Let the variable x represent the number. -The product of 4 and a number increased by 3 is 20.

Write the sentence as an equation. Let the variable x represent the number. -The sum of twice a number and 5 is 9.

Solve the equation. - $\frac { 6 } { 5 x } - \frac { 1 } { x + 1 } = \frac { 3 } { x ( 2 x + 2 ) }$