Question 1

Solve the problem.
-One side of a rectangle is 5 inches and the other side is x inches. What values of x will make the perimeter at most 36?

Accepted Answer

The perimeter of a rectangle is given by the formula $2l + 2w$, where $l$ and $w$ are the length and width of the rectangle, respectively. Substituting the given values, we get $2(5) + 2x \leq 36$. Simplifying, $10 + 2x \leq 36$, then $2x \leq 26$, and finally $x \leq 13$. Since $x$ represents the length of a side, it must be greater than 0, hence $0 < x \leq 13$.

Question 2

Solve the problem.
-One side of a triangle is 4 cm shorter than the base, x. The other side is 6 cm longer than the base. What lengths of the base will allow the perimeter of the triangle to be at least 44 cm?

Accepted Answer

The perimeter P of the triangle is given by $P = x + (x - 4) + (x + 6)$. Simplifying, we get $P = 3x + 2$. For the perimeter to be at least 44 cm, we set up the inequality $3x + 2 \geq 44$, solving for x gives $x \geq 14$.

Question 3

Solve the problem.
-A shop keeper is making a triangular sign for his store front, but he must keep the sign under 20 ft2 to adhere to zoning laws. If the base of the sign is 16 ft, what is the maximum height of the triangular sign?

Accepted Answer

The area of a triangle is given by the formula $ \frac{1}{2} \times \text{base} \times \text{height} $. To keep the area under 20 ft$^2$, and given the base is 16 ft, we solve for height in $ 20 = \frac{1}{2} \times 16 \times \text{height} $, which gives a maximum height of 2.5 ft.

Question 4

Solve the problem.
-One side of a rectangle is 16 inches and the other side is x inches. What values of x will make the perimeter at least 48?

Accepted Answer

The perimeter of a rectangle is given by $2(l + w)$, where $l$ and $w$ are the length and width of the rectangle, respectively. Substituting the given values, we get $2(16 + x) \geq 48$. Simplifying, $32 + 2x \geq 48$, and further $2x \geq 16$, which gives $x \geq 8$.

Question 5

Provide an appropriate response.
-Write the steps you would use to solve this equation: 4(x - 1)+ 7x = -2x.

Accepted Answer

The answer of Provide an appropriate response.
-Write the steps you...

Question 6

Solve the problem.
-One side of a rectangle is 13 inches and the other side is x inches. Find the value of x if the area must be at least 130 square inches.

Accepted Answer

The area of a rectangle is given by the formula length $\times$ width. Here, one side is 13 inches, and the other side is $x$ inches. To find $x$ when the area is at least 130 square inches, we set up the inequality $13x \geq 130$. Solving for $x$ gives $x \geq 10$, meaning $x$ must be greater than or equal to 10 inches to satisfy the condition.

Question 7

Provide an appropriate response.
-What value of K makes this equation equivalent to x = 3?
3x - 9 = K

Accepted Answer

The answer of Provide an appropriate response.
-What value of K...

Question 8

Solve the problem.
-The color guard is making new triangular flags that must have a base of 18 inches to fit on their flagpoles. What is the maximum length of the triangular flags, if they want to use a maximum of 297 in.2 of cloth?

Accepted Answer

The area of a triangle is given by (1/2) * base * height. With a base of 18 inches and a maximum area of 297 in², the height can be calculated as (2 * 297) / 18 = 33 inches.

Question 9

Provide an appropriate response.
-Express three consecutive integers, all in terms of x, if x is the largest integer.

Accepted Answer

The answer of Provide an appropriate response.
-Express three consecutive integers,...

Question 10

Provide an appropriate response.
-One number is twice another. If the larger number is m, how do you express the other number in terms of m?

Accepted Answer

The answer of Provide an appropriate response.
-One number is twice...

Question 11

Provide an appropriate response.
-What value of K makes this equation equivalent to x = 5?
5x + 12x - 2 = K + 4

Accepted Answer

The answer of Provide an appropriate response.
-What value of K...

Question 12

Solve the problem.
-One side of a rectangle is 2 times the other, and the perimeter is not to exceed 72. Find the possible values for x, the length of the shorter side.

Accepted Answer

The perimeter of a rectangle is given by $P = 2l + 2w$, where $l$ is the length and $w$ is the width. Given one side is twice the other, we can say $l = 2w$. Substituting $l$ with $2w$ in the perimeter formula gives $P = 2(2w) + 2w = 6w$. Since the perimeter must not exceed 72, we have $6w \leq 72$. Dividing both sides by 6 gives $w \leq 12$. Since $w$ represents the length of the shorter side and must be greater than 0, the possible values for $x$ are $0 < x \leq 12$.

Question 13

Provide an appropriate response.
-The solution of the equation 7y - 6 = 7y + 3 is zero.

Accepted Answer

It has no solution.

Question 14

Provide an appropriate response.
-What value of K makes this equation equivalent to x = 3? $$\frac { 9 } { K + x } = 3$$

Accepted Answer

The answer of Provide an appropriate response.
-What value of K...

Question 15

Solve the problem.
-If the formula R = -0.037t + 50.1 can be used to predict the world record in the 400-meter dash t years after 1925, for what years will the world records be 48.1 seconds or less?

Accepted Answer

The answer of Solve the problem.
-If the formula R =...

Question 16

Solve the problem.
-If the formula P = 0.5643Y - 1092.57 can be used to predict the average price of a theater ticket after 1945, for what years will the average theater ticket price be at least 42 dollars? (Y is the actual year.)

Accepted Answer

To find the years when the average theater ticket price will be at least $42, we set the equation $P = 0.5643Y - 1092.57 \geq 42$. Solving for $Y$, we get $Y \geq 2011$. Therefore, the correct answer is 2011 or after.

Question 17

Solve the problem.
-The area of a triangle must be at most 52 square inches, the base is 13 inches, and the height is x inches. Find the possible values for x.

Accepted Answer

The area of a triangle is given by the formula $ \frac{1}{2} \times \text{base} \times \text{height} $. Given that the area must be at most 52 square inches and the base is 13 inches, we can set up the inequality $ \frac{1}{2} \times 13 \times x \leq 52 $. Simplifying, we get $ \frac{13x}{2} \leq 52 $, which further simplifies to $ x \leq 8 $. Since the height of a triangle must be greater than 0, the possible values for x are $0 < x \leq 8$.

Question 18

Provide an appropriate response.
-Find all values of x that make this statement true: (x - 6)- 7 = (x - 7)- 6.

Accepted Answer

The answer of Provide an appropriate response.
-Find all values of...

Question 19

Provide an appropriate response.
-The solution for the equation $ 4 ( 9 \mathrm {~s} - 5 ) = 36 \mathrm {~s} - 20$ is given as  0 . Is this correct? Explain.

Accepted Answer

The solution is all real numbers. Explanations will vary.

Question 20

Provide an appropriate response.
-Find all values of s that make this statement true: $$4 ( 8 s - 6 ) = 32 s - 24$$

Accepted Answer

The answer of Provide an appropriate response.
-Find all values of...

Quiz 2: Solving Equations and Inequalities

Solve the problem. -One side of a rectangle is 5 inches and the other side is x inches. What values of x will make the perimeter at most 36?

Solve the problem. -One side of a triangle is 4 cm shorter than the base, x. The other side is 6 cm longer than the base. What lengths of the base will allow the perimeter of the triangle to be at least 44 cm?

Solve the problem. -A shop keeper is making a triangular sign for his store front, but he must keep the sign under 20 ft2 to adhere to zoning laws. If the base of the sign is 16 ft, what is the maximum height of the triangular sign?

Solve the problem. -One side of a rectangle is 16 inches and the other side is x inches. What values of x will make the perimeter at least 48?

Provide an appropriate response. -Write the steps you would use to solve this equation: 4(x - 1)+ 7x = -2x.

Solve the problem. -One side of a rectangle is 13 inches and the other side is x inches. Find the value of x if the area must be at least 130 square inches.

Provide an appropriate response. -What value of K makes this equation equivalent to x = 3? 3x - 9 = K

Solve the problem. -The color guard is making new triangular flags that must have a base of 18 inches to fit on their flagpoles. What is the maximum length of the triangular flags, if they want to use a maximum of 297 in.2 of cloth?

Provide an appropriate response. -Express three consecutive integers, all in terms of x, if x is the largest integer.

Provide an appropriate response. -One number is twice another. If the larger number is m, how do you express the other number in terms of m?

Provide an appropriate response. -What value of K makes this equation equivalent to x = 5? 5x + 12x - 2 = K + 4

Solve the problem. -One side of a rectangle is 2 times the other, and the perimeter is not to exceed 72. Find the possible values for x, the length of the shorter side.

Provide an appropriate response. -The solution of the equation 7y - 6 = 7y + 3 is zero.

Provide an appropriate response. -What value of K makes this equation equivalent to x = 3? $\frac { 9 } { K + x } = 3$

Solve the problem. -If the formula R = -0.037t + 50.1 can be used to predict the world record in the 400-meter dash t years after 1925, for what years will the world records be 48.1 seconds or less?

Solve the problem. -If the formula P = 0.5643Y - 1092.57 can be used to predict the average price of a theater ticket after 1945, for what years will the average theater ticket price be at least 42 dollars? (Y is the actual year.)

Solve the problem. -The area of a triangle must be at most 52 square inches, the base is 13 inches, and the height is x inches. Find the possible values for x.

Provide an appropriate response. -Find all values of x that make this statement true: (x - 6)- 7 = (x - 7)- 6.

Provide an appropriate response. -The solution for the equation $4 ( 9 \mathrm {~s} - 5 ) = 36 \mathrm {~s} - 20$ is given as 0 . Is this correct? Explain.

Provide an appropriate response. -Find all values of s that make this statement true: $4 ( 8 s - 6 ) = 32 s - 24$

Quiz 2: Solving Equations and Inequalities

Solve the problem. -One side of a rectangle is 5 inches and the other side is x inches. What values of x will make the perimeter at most 36?

Solve the problem. -One side of a triangle is 4 cm shorter than the base, x. The other side is 6 cm longer than the base. What lengths of the base will allow the perimeter of the triangle to be at least 44 cm?

Solve the problem. -A shop keeper is making a triangular sign for his store front, but he must keep the sign under 20 ft2 to adhere to zoning laws. If the base of the sign is 16 ft, what is the maximum height of the triangular sign?

Solve the problem. -One side of a rectangle is 16 inches and the other side is x inches. What values of x will make the perimeter at least 48?

Provide an appropriate response. -Write the steps you would use to solve this equation: 4(x - 1)+ 7x = -2x.

Solve the problem. -One side of a rectangle is 13 inches and the other side is x inches. Find the value of x if the area must be at least 130 square inches.

Provide an appropriate response. -What value of K makes this equation equivalent to x = 3? 3x - 9 = K

Solve the problem. -The color guard is making new triangular flags that must have a base of 18 inches to fit on their flagpoles. What is the maximum length of the triangular flags, if they want to use a maximum of 297 in.2 of cloth?

Provide an appropriate response. -Express three consecutive integers, all in terms of x, if x is the largest integer.

Provide an appropriate response. -One number is twice another. If the larger number is m, how do you express the other number in terms of m?

Provide an appropriate response. -What value of K makes this equation equivalent to x = 5? 5x + 12x - 2 = K + 4

Solve the problem. -One side of a rectangle is 2 times the other, and the perimeter is not to exceed 72. Find the possible values for x, the length of the shorter side.

Provide an appropriate response. -The solution of the equation 7y - 6 = 7y + 3 is zero.

Provide an appropriate response. -What value of K makes this equation equivalent to x = 3? 9K+x=3\frac { 9 } { K + x } = 3K+x9​=3

Solve the problem. -If the formula R = -0.037t + 50.1 can be used to predict the world record in the 400-meter dash t years after 1925, for what years will the world records be 48.1 seconds or less?

Solve the problem. -If the formula P = 0.5643Y - 1092.57 can be used to predict the average price of a theater ticket after 1945, for what years will the average theater ticket price be at least 42 dollars? (Y is the actual year.)

Solve the problem. -The area of a triangle must be at most 52 square inches, the base is 13 inches, and the height is x inches. Find the possible values for x.

Provide an appropriate response. -Find all values of x that make this statement true: (x - 6)- 7 = (x - 7)- 6.

Provide an appropriate response. -The solution for the equation 4(9 s−5)=36 s−20 4 ( 9 \mathrm {~s} - 5 ) = 36 \mathrm {~s} - 204(9 s−5)=36 s−20 is given as 0 . Is this correct? Explain.

Provide an appropriate response. -Find all values of s that make this statement true: 4(8s−6)=32s−244 ( 8 s - 6 ) = 32 s - 244(8s−6)=32s−24

Provide an appropriate response. -What value of K makes this equation equivalent to x = 3? $\frac { 9 } { K + x } = 3$

Provide an appropriate response. -The solution for the equation $4 ( 9 \mathrm {~s} - 5 ) = 36 \mathrm {~s} - 20$ is given as 0 . Is this correct? Explain.

Provide an appropriate response. -Find all values of s that make this statement true: $4 ( 8 s - 6 ) = 32 s - 24$