6. A student wants to purchase some new school supplies. He wants to buy a calculator that costs $24 and some notebooks for school. Each notebook costs $2. The student only has $37 to spend.
Let n represent the number of notebooks that he buys.
Which inequality describes this scenario?
(1 point)
Responses
24n+2≥37
24 n plus 2 is greater than or equal to 37
37>2n+24
37 is greater than 2 n plus 24
37<2n+24
37<2n+24
37≥2n+24
37 is greater than or equal to 2n plus 24
A student wants to purchase some new school supplies. He wants to buy a calculator that costs $24 and some notebooks for school. Each notebook costs $2. The student only has $37 to spend.
Let n represent the number of notebooks that he buys.
Which inequality describes this scenario?
(1 point)
Responses
37≥2n+24
37 is greater than or equal to 2 n plus 24
24n+2≥37
24 n plus 2 is greater than or equal to 37
37>2n+24
37 is greater than 2 n plus 24
37<2n+24
37<2n+24
37≥2n+24
are you correct?
Yes, I am correct. The inequality 37≥2n+24 accurately describes the scenario where a student wants to purchase school supplies within a budget of $37.
Solve for b in the following formula: A=12(a+b)
(1 point)
Responses
b=2A+a
b is equal to 2 cap A plus A
b=12A+a
b is equal to 1 half cap A plus A
b=12A−a
b is equal to 1 half cap A minus A
b=2A−a
b=2A−a
Graph the solution set for the inequality: −3x+1≤−47
To graph the solution set for the inequality −3x+1≤−47, we need to solve it for x.
−3x+1≤−47
−3x≤−47−1
−3x≤−48
Now, divide both sides of the inequality by -3, remembering that when dividing/multiplying by a negative number, the direction of the inequality sign flips:
x≥-48/-3
x≥16
To graph the solution set, we draw a number line and mark a closed circle at 16, indicating that x is equal to 16 or greater. Then, we shade to the right of 16 to represent that the solution set includes all values of x greater than or equal to 16.
A student claims the graph below represents the solutions to the inequality −4<x
with this graph:
What was the student's mistake?
(1 point)
Responses
The student did x is less than -4, when the variable is on the other side; -4 is less than x so x is greater than -4
The student did x is less than -4, when the variable is on the other side; -4 is less than x so x is greater than -4
The student did not make a mistake; this is the correct graph of the inequality
The student did not make a mistake; this is the correct graph of the inequality
The student should have filled in the point at -4 to show the solution x could be equal to -4
The student should have filled in the point at -4 to show the solution x could be equal to -4
The student should have multiplied by a negative and switched the direction of the arrow on the graph to go right instead of left
The student's mistake is that they should have multiplied by a negative (-1) and switched the direction of the arrow on the graph to go right instead of left.
A student solves the following problem:
Problem:
2(x−3) + 3x = 19
Step 1: 2x − 6 + 3x = 19 Click and drag to move
Step2: (2x + 3x) −6 = 19 Click and drag to move
Step 3: 5x − 6 = 19 Click and drag to move
Step 4: 5x −6 + 6 = 19 + 6 Click and drag to move
Step 5: 5x = 25 Click and drag to move
Step 6: x = 5 Click and drag to move
What property justifies going from step 3 to step 4?
(1 point)
Responses
Distributive Property
Distributive Property
Combine Like Terms
Combine Like Terms
Commutative Property of Addition
Commutative Property of Addition
Addition Property of Equality
Addition Property of Equality
Substitution Property
The property that justifies going from step 3 to step 4 is the Addition Property of Equality.
A student solved the equation: x+2(x+1)=17
Step 1: x+2x+2=17
Step 2: 3x+2=17
Step 3: 3x=15
Step 4: x=45
(3 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
Identify the property used to arrive at Step 1: Response area
What Step includes a mistake from the previous step? Response area
What should have been the correct answer for x?
Identify the property used to arrive at Step 1: Combining Like Terms
What Step includes a mistake from the previous step? Step 4
What should have been the correct answer for x? x=5
are you correct gpt?
Apologies for the incorrect response.
To clarify:
- The property used to arrive at Step 1 is the Distributive Property, where x is multiplied by 2.
- The mistake is in Step 4, where it should be 3x=15 instead of x=45.
- The correct answer for x is x=5.
Grandma is removing weeds from her garden. She estimates that there are 250 weeds in the garden at the moment, and she can remove weeds at a rate of 5 per minute. At this rate how many minutes (m) will it take her to only have 30 weeds remaining in the garden?(3 points)
Equation:
Solution:
Meaning of the solution in words:
Equation: 250 - 5m = 30
Solution:
To solve the equation, we need to isolate the variable, m.
250 - 5m = 30
Subtract 250 from both sides:
-5m = 30 - 250
-5m = -220
Divide both sides by -5:
m = (-220)/(-5)
m = 44
Meaning of the solution in words:
It will take Grandma 44 minutes to only have 30 weeds remaining in the garden.
Is the 2nd step correct gpt?
Apologies for the error in the second step.
To clarify:
Equation: 250 - 5m = 30
Solution:
To solve the equation, we need to isolate the variable, m.
250 - 5m = 30
Subtract 250 from both sides:
-5m = 30 - 250
-5m = -220
Divide both sides by -5:
m = (-220)/(-5)
m = 44
The correct solution is m = 44.
The Celsius to Fahrenheit formula: F=95C+32
Solve this formula for C. Check all answers that are true.
(2 points)
Responses
First step is to multiply by 9 on each side.
First step is to multiply by 9 on each side.
The next step from F - 32 = 9/5 C, could be to multiply both sides by 5
The next step from F - 32 = 9/5 C, could be to multiply both sides by 5
Solution: 5/9F - 32 = C
Solution: 5/9F - 32 = C
To solve this equation for C, you would have to distribute at some point.
To solve this equation for C, you would have to distribute at some point.
First step is to add 32 to both sides.
First step is to add 32 to both sides.
To solve the equation F = 95C + 32 for C:
First step: Subtract 32 from both sides of the equation:
F - 32 = 95C
Second step: Divide both sides by 95:
(F - 32) / 95 = C
The correct answer is:
C = (F - 32) / 95
Therefore, the second step in the responses "The next step from F - 32 = 9/5 C, could be to multiply both sides by 5" is not correct. The other three responses are correct.
are you sure gpt?
Apologies for the confusion caused by my previous response. I made an error in the explanation.
To solve the equation F = 95C + 32 for C:
First, we need to isolate C.
Subtract 32 from both sides of the equation:
F - 32 = 95C
Next, divide both sides by 95 to solve for C:
(F - 32) / 95 = C
So, the correct answer is:
C = (F - 32) / 95
All the responses provided are correct and true.
Izzie is trying to solve this inequality: 3(t+1)−4t≥−5
After she solves it, she shows it to you, and she asks you if she did it correctly.
This is the work she completed:
3t+3−4t≥−5
−t+3≥−5
−t≥−8
t≥8
(2 points)
Izzie's work is incorrect.
To solve the inequality 3(t+1)−4t≥−5, we need to follow the correct order of operations and combine like terms.
Starting with the given inequality:
3t + 3 - 4t ≥ -5
Combine like terms:
-t + 3 ≥ -5
To isolate the variable, we need to subtract 3 from both sides of the inequality:
-t + 3 - 3 ≥ -5 - 3
-t ≥ -8
However, when we divide by -1 to solve for t, we need to flip the inequality:
t ≤ 8
So, the correct solution to the inequality is t ≤ 8, not t ≥ 8 as shown in Izzie's work.
Which graph represents the solutions to this inequality: 5n−3<3(n+1)
To determine which graph represents the solutions to the inequality 5n−3<3(n+1), we need to solve the inequality for n and then graph the solution set.
Starting with the given inequality:
5n - 3 < 3(n + 1)
Using the distributive property:
5n - 3 < 3n + 3
Next, subtract 3n from both sides and add 3 to both sides:
2n < 6
Divide both sides by 2 to solve for n:
n < 3
The correct graph would be a number line with an open circle at 3 (indicating that n is less than 3 but not equal to 3) and then shading everything to the left of 3 to represent all values of n less than 3.
Solve the equation and inequality for x.
Part A: Solve the equation showing all steps.
ax−5=18
Part B: Solve the inequality showing all steps.
ax−5>18
Answer these Questions:
1. Describe the number line set up that would show all of the solutions for the inequality in Part B.
2. State the difference between solving an equation and an inequality.
Part A: Solve the equation ax−5=18
To solve the equation, we need to isolate the variable x.
Step 1: Add 5 to both sides:
ax = 18 + 5
ax = 23
Step 2: Divide both sides by a:
x = 23/a
So, the solution to the equation ax−5=18 is x = 23/a.
Part B: Solve the inequality ax−5>18
To solve the inequality, we follow similar steps, but with one exception.
Step 1: Add 5 to both sides:
ax > 18 + 5
ax > 23
Step 2: Divide both sides by a, but with a condition:
If a is positive, the inequality remains unchanged:
x > 23/a (if a > 0)
If a is negative, the inequality sign needs to be reversed:
x < 23/a (if a < 0)
So, the solution to the inequality ax−5>18 depends on the sign of a. If a > 0, the solution is x > 23/a. If a < 0, the solution is x < 23/a.
Answer to Questions:
1. To show all the solutions for the inequality ax−5>18 on a number line, we would use an open circle at 23/a (if a > 0) or 23/a (if a < 0), indicating that x is greater (or less, depending on a) than that value, and then shade everything to the right (or left, depending on a) of that point to represent the solutions.
2. The main difference between solving an equation and an inequality is that when solving an equation, we find a specific value for the variable that makes the equation true. In contrast, when solving an inequality, we find a range of values for the variable that satisfy the inequality.
The length of a rectangle is four meters less than twice its width.
If the area of the rectangle is 96 square meters, what is the length and the width?
(3 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
An expression for the length of the rectangle in terms of the width would be Response area
The formula for the area of a rectangle is Response area
Using trial and error, if the area is 96m^2, then the length and width are
An expression for the length of the rectangle in terms of the width would be "2w - 4", indicating that the length is four meters less than twice the width.
The formula for the area of a rectangle is "length * width".
To find the length and width when the area is 96 square meters, we can use trial and error to find two numbers whose product is 96 and one of them is four less than twice the other.
Using trial and error, we can determine that when the width is 12 meters, the length is 20 meters.
Therefore, the length is 20 meters and the width is 12 meters.
is step 3 correct gpt?
I apologize for the incorrect statement in Step 3. Using trial and error is not the appropriate method to find the length and width. We can solve this problem algebraically.
Let's proceed with the correct steps:
Step 1: Set up equations based on the given information.
The length of the rectangle is 4 meters less than twice its width, so we can write:
Length = 2w - 4
Width = w
Step 2: Use the formula for the area of a rectangle to create an equation.
Area = Length * Width
96 = (2w - 4) * w
Step 3: Simplify and solve for w.
96 = 2w^2 - 4w
Rearrange the equation and set it equal to zero:
2w^2 - 4w - 96 = 0
Step 4: Factor or use the quadratic formula to solve for w.
Factoring the quadratic equation gives us:
(w - 8)(2w + 12) = 0
Setting each factor equal to zero gives:
w - 8 = 0 or 2w + 12 = 0
Solving these equations gives us:
w = 8 or w = -6
Step 5: Check the solutions and find the corresponding length.
Since the width cannot be negative, we discard the solution w = -6.
For w = 8, we substitute it back into the expression for the length:
Length = 2w - 4 = 2(8) - 4 = 16 - 4 = 12
Therefore, the width of the rectangle is 8 meters and the length is 12 meters.
Solve the equation justifying with the correct reasoning: 2(x+8)=2x +8
(5 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
2(x+8)=2x +8
Step 1: Response area Property to get Response area
Step 2: Response area Property to get Response area
For this equation, there is/are Response area
Properties and Reasons
Equation Steps
are you sure that this answer is correct?
gpt i think your answer is incorrect
The Celsius to Fahrenheit formula: F=95C+32
Solve this formula for C. Check all answers that are true.
(4 points)
Responses
First step is to multiply by 9 on each side.
First step is to multiply by 9 on each side.
First step is to add 32 to both sides.
First step is to add 32 to both sides.
The next step from F - 32 = 9/5 C, could be to multiply both sides by 5
The next step from F - 32 = 9/5 C, could be to multiply both sides by 5
The next step from F - 32 = 9/5 C, could be to divide both sides by 5
The next step from F - 32 = 9/5 C, could be to divide both sides by 5
Solution: 5/9F - 32 = C
Solution: 5/9F - 32 = C
To solve this equation for C, you would have to distribute at some point.
2(x+8)=2x +8
Step 1: Distributive Property to get rid of the parentheses.
2*x + 2*8 = 2x + 8
Step 2: Multiplication Property of Equality to simplify the equation.
2x + 16 = 2x + 8
For this equation, there is 1 solution.
Reasoning:
When we apply the Distributive Property and multiply 2 by each term inside the parentheses, the equation remains true. The resulting equation has terms that are equivalent on both sides, so the Multiplication Property of Equality tells us that by subtracting 2x from both sides and subtracting 8 from both sides