Question

Consider the following expressions:

6 - 1/4 times 28 - 2 1/4 ________ 3/4 + 2 times 2 - 11
Which operator cab be inserted to make the statement true?

Answer 1

[AI-generated answer — Bot GPT 3.5]

The operator that can be inserted to make the statement true is the addition operator (+). The correct expression would be:

6 - 1/4 * 28 - 2 1/4 + 3/4 + 2 * 2 - 11

Answer 2

Consider the following expressions:

6 - 1/4 times 28 - 2 1/4 ________ 3/4 + 2 times 2 - 11
Which operator cab be inserted to make the statement true?
A. >_
B. =
C. <
D. >

Answer 3

[AI-generated answer — Bot GPT 3.5]

To determine which operator can be inserted to make the statement true, let's simplify the given expression:

6 - 1/4 * 28 - 2 1/4 ________ 3/4 + 2 * 2 - 11
= 6 - 7 - 2 1/4 ________ 3/4 + 4 - 11
= -1 - 2 1/4 ________ 3/4 - 7
= -1 - (2 + 1/4) ________ (3 - 7/4)
= -1 - (9/4) ________ (12/4 - 7/4)
= -1 - (9/4) ________ 5/4

Now let's compare the two sides of the expression:
-1 - (9/4) _____ 5/4

To evaluate this comparison, we can convert both sides to have a common denominator:
-1 - (9/4) _____ 5/4
-4/4 - (9/4) _____ 5/4
(-4 - 9)/4 _____ 5/4
-13/4 _____ 5/4

Since -13/4 is less than 5/4, we can conclude that the correct operator to make the statement true is C. < (less than).

Answer 4

Which of the following values for b makes the inequality -32.5 - 6b >_ 48.5 true?

Answer 5

[AI-generated answer — Bot GPT 3.5]

To find the values of b that make the inequality true, we can solve for b.

The inequality is: -32.5 - 6b ≥ 48.5

Let's start by isolating the variable b.

Add 32.5 to both sides of the inequality:
-32.5 - 32.5 - 6b ≥ 48.5 + 32.5
-6b ≥ 81

Divide both sides of the inequality by -6, remembering to reverse the inequality sign when dividing by a negative number:
b ≤ 81 / -6

Simplifying:
b ≤ -13.5

Therefore, any value of b that is less than or equal to -13.5 will make the inequality -32.5 - 6b ≥ 48.5 true.

Answer 6

Which of the following equations has the solution set {-3}?

A. 4x + 8 = -4
B. 4x + 8 = 4
C. 4x - 8 = 4
D. 4x - 8 = -4

Answer 7

[AI-generated answer — Bot GPT 3.5]

To find which equation has the solution set {-3}, we can substitute -3 for x and see which equation holds true.

A. 4(-3) + 8 = -4
-12 + 8 = -4
-4 = -4 - This equation is false.

B. 4(-3) + 8 = 4
-12 + 8 = 4
-4 = 4 - This equation is false.

C. 4(-3) - 8 = 4
-12 - 8 = 4
-20 = 4 - This equation is false.

D. 4(-3) - 8 = -4
-12 - 8 = -4
-20 = -4 - This equation is true.

Therefore, the equation that has the solution set {-3} is D. 4x - 8 = -4.

Answer 8

Which set expresses the solution to the compound inequality -2 <_ 2x + 1/3 -4 <1?

A. { x : 5/2 <_ x < 7}
B. { x : 5/2 <_ x < 1}
C. { x : 3 <_ x <7}
D. { x : -3/2 <_ x <1}

Answer 9

[AI-generated answer — Bot GPT 3.5]

To find the solution to the compound inequality -2 ≤ 2x + 1/3 - 4 < 1, we need to solve each individual inequality and then find the intersection of the solution sets.

First, let's solve the left inequality: -2 ≤ 2x + 1/3 - 4

Adding 4 to both sides:
2 ≤ 2x + 1/3

Subtracting 1/3 from both sides:
2 - 1/3 ≤ 2x

Multiplying both sides by 1/2:
1 2/3 ≤ x

So the solution to the left inequality is x ≥ 1 2/3 or x ≥ 5/3.

Now let's solve the right inequality: 2x + 1/3 - 4 < 1

Adding 4 to both sides:
2x + 1/3 - 4 + 4 < 1 + 4
2x + 1/3 < 5

Subtracting 1/3 from both sides:
2x < 5 - 1/3

Simplifying:
2x < 14/3

Dividing both sides by 2:
x < 7/3

So the solution to the right inequality is x < 7/3.

As we are looking for the intersection of both solution sets, the solution to the compound inequality is: x ≥ 5/3 and x < 7/3.

Simplifying this interval notation, we get:
{ x: 5/3 ≤ x < 7/3}
which is the same as:
{ x: 1 2/3 ≤ x < 2 1/3}

Therefore, the correct answer is A. { x: 5/2 ≤ x < 7}

Answer 10

Which linear equation has the same solution set as 9d/4 + 3 - 2d = -2 ( 3 + d)?

Answer 11

[AI-generated answer — Bot GPT 3.5]

To find the linear equation that has the same solution set as the given equation, we need to simplify the given equation and convert it into the standard form of a linear equation.

9d/4 + 3 - 2d = -2(3 + d)

First, distribute -2 to both terms inside the parentheses:
9d/4 + 3 - 2d = -6 - 2d

Next, let's combine like terms on both sides of the equation:
9d/4 - 2d + 2d = -6 - 3

Simplifying further:
9d/4 = -9

To eliminate the fraction, we can multiply both sides of the equation by 4:
4 * (9d/4) = 4 * (-9)
9d = -36

Finally, divide both sides of the equation by 9 to solve for d:
d = -36/9
d = -4

Thus, the linear equation that has the same solution set as 9d/4 + 3 - 2d = -2(3 + d) is d = -4.

Answer 12

Which linear equation has the same solution set as 9d/4 + 3 - 2d = -2 ( 3 + d)?

A. 35 - 15b = 95
B. 4w - 2 ( 7 - 3w) = w/3 - 6
C. 14 = 7/4 x - 19/4 + 2x
D. 3x/5 - 10 = -7

Answer 13

[AI-generated answer — Bot GPT 3.5]

To find the linear equation that has the same solution set as the given equation, we need to simplify the given equation and compare it with the options provided.

9d/4 + 3 - 2d = -2(3 + d)

First, distribute -2 to the terms inside the parentheses:
9d/4 + 3 - 2d = -6 - 2d

Next, let's combine like terms on both sides of the equation:
9d/4 - 2d + 2d = -6 - 3

Simplifying further:
9d/4 = -9

To eliminate the fraction, we can multiply both sides of the equation by 4:
4 * (9d/4) = 4 * (-9)
9d = -36

Finally, divide both sides of the equation by 9 to solve for d:
d = -36/9
d = -4

Now let's compare the simplified equation d = -4 with the options provided:

A. 35 - 15b = 95
B. 4w - 2 ( 7 - 3w) = w/3 - 6
C. 14 = 7/4 x - 19/4 + 2x
D. 3x/5 - 10 = -7

None of the options provided match the solution d = -4.

Therefore, none of the options A, B, C, or D have the same solution set as the given equation 9d/4 + 3 - 2d = -2(3 + d).

Answer 14

Which option is the solution to the inequality -12g + 18 > -5g + 53?

A. g > -5
B. g < -5
C. g < 71/-17
D. g > 71/-17

Answer 15

[AI-generated answer — Bot GPT 3.5]

To find the solution to the inequality -12g + 18 > -5g + 53, we need to solve for g.

Let's start by getting all the g terms on one side of the inequality:

-12g + 18 > -5g + 53

Add 12g and subtract 53 from both sides:
-12g + 12g + 18 - 53 > -5g + 12g + 53 - 53
18 - 53 > -5g + 12g

Combine like terms:
-35 > 7g

Divide both sides by 7, remembering to flip the direction of the inequality when dividing by a negative number:
-35/7 < g

Simplify:
-5 < g

So, the solution to the inequality -12g + 18 > -5g + 53 is g > -5.

Therefore, the correct option is A. g > -5.

Answer 16

Find the solution set of 4x - 3 > 9 and x + 6 > -2.

Answer 17

[AI-generated answer — Bot GPT 3.5]

To find the solution set of the inequalities 4x - 3 > 9 and x + 6 > -2, we need to solve each inequality separately and then find the intersection of the solution sets.

Let's start with the first inequality:
4x - 3 > 9

Add 3 to both sides:
4x - 3 + 3 > 9 + 3
4x > 12

Divide both sides by 4:
x > 12/4
x > 3

So, the solution to the first inequality is x > 3.

Now let's solve the second inequality:
x + 6 > -2

Subtract 6 from both sides:
x + 6 - 6 > -2 - 6
x > -8

So, the solution to the second inequality is x > -8.

To find the intersection of the solution sets, we take the smaller of the two values. In this case, the smaller value is x > 3.

Therefore, the solution set for the given inequalities is { x : x > 3 }.

Answer 18

What is the undefined value for the equation 14 = 7x/9+4x?

Answer 19

[AI-generated answer — Bot GPT 3.5]

To find the undefined value for the equation 14 = 7x/9 + 4x, we need to determine if there are any values of x that would make the equation undefined.

The equation is defined for all values of x except for those that would make the denominator of the expression 7x/9 equal to zero.

To find the undefined value, we set the denominator equal to zero and solve for x:

9 = 0

However, there is no value of x that satisfies this equation. Therefore, there is no undefined value for the given equation 14 = 7x/9 + 4x. The equation is defined for all real values of x.

Answer 20

What is the undefined value for the equation 14 = 7x/9+4x?

A. x not equal 9/4
B. x not equal 0
C. x not equal -9/4
D. x not equal -4/9

Answer 21

[AI-generated answer — Bot GPT 3.5]

To find the undefined value for the equation 14 = 7x/9 + 4x, we need to determine if there are any values of x that would make the equation undefined.

The equation is defined for all values of x except for those that would make the denominator equal to zero. In this case, the denominator is the sum of 7x/9 and 4x.

To find when this denominator would be zero, we set it equal to zero and solve for x:

7x/9 + 4x = 0

To simplify, we can find a common denominator:
(7x + 36x)/9 = 0

Combining like terms:
43x/9 = 0

Now, to have a zero denominator, we need 43x to be equal to zero:
43x = 0

Dividing both sides by 43, we find:
x = 0/43
x = 0

Therefore, the undefined value for the equation 14 = 7x/9 + 4x is B. x is not equal to 0.

Answer 22

Solve for s in the following equation: 11/s+3 + 4 = 5

Answer 23

[AI-generated answer — Bot GPT 3.5]

To solve for s in the equation 11/s + 3 + 4 = 5, we follow these steps:

1. First, let's simplify the equation by combining like terms:
11/s + 7 = 5

2. Next, we subtract 7 from both sides of the equation:
11/s = 5 - 7
11/s = -2

3. To isolate the variable s, we multiply both sides of the equation by s:
s * (11/s) = s * (-2)

4. The s term in the numerator and denominator cancels out on the left side of the equation:
11 = -2s

5. Lastly, we divide both sides by -2 to solve for s:
11 / -2 = -2s / -2
-5.5 = s

Therefore, the solution to the equation 11/s + 3 + 4 = 5 is s = -5.5.

Answer 24

Which correctly shows the area formula for a trapezoid, A = 1/2 h ( b1 + b2), rearranged for the quanity of interest h?

Answer 25

[AI-generated answer — Bot GPT 3.5]

To rearrange the area formula for a trapezoid, A = 1/2 h ( b1 + b2 ), for the height h, the following step can be followed:

1. Start by multiplying both sides of the equation by 2 to eliminate the fraction:
2A = h ( b1 + b2 )

2. Divide both sides of the equation by ( b1 + b2 ) to isolate h:
2A / ( b1 + b2 ) = h

Therefore, the correctly rearranged formula for the height (h) of a trapezoid is:
h = 2A / ( b1 + b2 )

Answer 26

Given the formula for time in terms of the distance and velocity, t = d/v, what operation would you use to isolate the distance?

Answer 27

[AI-generated answer — Bot GPT 3.5]

To isolate the distance (d) in the formula t = d/v, you would use multiplication.

To isolate d, multiply both sides of the equation by v:

t * v = (d/v) * v

This simplifies to:

t * v = d * (v/v)

Since v/v equals 1, we have:

t * v = d

Now, d is isolated and we have the formula for distance:

d = t * v

Answer 28

Given the formula for time in terms of the distance and velocity, t = d/v, what operation would you use to isolate the distance?

A. addition
B. subtraction
C. multiplication
D. division

Answer 29

[AI-generated answer — Bot GPT 3.5]

To isolate the distance (d) in the formula t = d/v, you would use multiplication.

By multiplying both sides of the equation by v, we have:

t * v = (d/v) * v

This simplifies to:

t * v = d

Therefore, the correct operation to isolate the distance is C. multiplication.

Answer 30

Sebastian needed to buy new tires for his car. The tire shop was running a holiday deal that allowed customers to buy three tires and get the fourth tire for $5. If the total price for all his tires was $237.50, what was the cost of each of the first three tires?

Answer 31

[AI-generated answer — Bot GPT 3.5]

Let's assume the cost of each of the first three tires is x dollars.

According to the deal, if Sebastian buys three tires at the cost of x dollars each, the fourth tire would cost $5.

So, the total cost of four tires would be 3x + $5.

Given that the total price of all four tires is $237.50, we can set up the equation:

3x + $5 = $237.50

Subtracting $5 from both sides of the equation:

3x = $237.50 - $5
3x = $232.50

To find the cost of each of the first three tires (x), we divide both sides of the equation by 3:

x = $232.50 / 3
x ≈ $77.50

Therefore, the cost of each of the first three tires is approximately $77.50.

Answer 32

A 10-gallon water cooler in an office provides water for the whole department. Each hour, 30 ounces of water are removed from the cooler and drunk by office workers. Write an equation to show how long the water in the cooler will last. (10 gallons is 1,280 ounces.)

A. 1,280 + 30h = 0
B. 30h = 10
C. 1,280 - 30h = 0
D. 10 - 30h = 0

Answer 33

A 10-gallon water cooler in an office provides water for the whole department. Each hour, 30 ounces of water are removed from the cooler and drunk by office workers. Write an equation to show how long the water in the cooler will last. (10 gallons is 1,280 ounces.)

A. 1,280 + 30h = 0
B. 30h = 10
C. 1,280 - 30h = 0
D. 10 - 30h = 0