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1. Which equation has the same solution for x as this equation: x−12=40

?(1 point)
Responses

x4=13
x over 4 is equal to 13

x+12=−40
x plus 12 is equal to negative 40

x2=14
x over 2 is equal to 14

12−x=40
12 minus x is equal to 40
Question 3
2. The length of a rectangle is four meters less than twice its width.

If the area of the rectangle is 96 m^2, 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 96 m^2, then the length and width are Response area

Question 4
3. Match the equation with its solution(s).(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.
3i+5=3i+7
3i+2i−7=18
3i+5=2i−7
3(2i+7)=6i+21
3(2i−7)=15
Question 5
4. Solve the equation justifying each step with the correct reasoning.

2(x+8)=2x+8
(5 points)
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Step 1: Response area Property to get Response area simplified equation

Step 2: Response area Property to get Response area simplified equation

For this equation, there is/are Response area

Properties and Reasons
Equation simplified
Question 6
5. Match the description of the one variable equation with the number of solutions it will have.(4 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.
x−7=7−x
3(x+5)=3x+5
10−x=25
2(x+4)=2x+5+3
Question 7
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 is greater than or equal to 2 n plus 24

37<2n+24
37<2n+24
Question 8
7. Solve for b in the following equation: A=12(a+b)
(1 point)
Responses

b=2A−a
b is equal to 2 cap A minus A

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
Question 9
8. Graph the solutions for the inequality: −3x+1≤−47
(2 points)
Responses

Question 10
9. A student claims that graph below represents the solutions to the inequality: −4<x

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 multiplied by a negative and switched the direction of the arrow on the graph to go right instead of left
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 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
Question 11
10. A student solves the following equation:

Problem: 2(x−3)+3x=19
Step 1: 2x−6+3x=19
Step 2: (2x+3x)−6=19
Step 3: 5x−6=19
Step 4: 5x−6+6=19+6
Step 5: 5x=25
Step 6: x=5
What property justifies going from step 3 to step 4?

(1 point)
Responses

Addition Property of Equality
Addition Property of Equality

Commutative Property of Addition
Commutative Property of Addition

Combine Like Terms
Combine Like Terms

Substitution Property
Substitution Property

Distributive Property
Distributive Property
Question 12
11. 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 made from the previous step? Response area

What should have been the correct answer for x ? Response area

Question 13
12. 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 will it take her to only have 30 weeds remaining in the garden?(3 points)
Equation:

Solution:

Meaning of the solution in words:

Question 14
13. 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.
To solve this equation for C, you would have to distribute at some point.
Question 15
14. Izzie is trying to solve this inequality: 3(t+1)−4t≥−5
After she solves it, she shows it to you and asks you if she did it correctly.

This is the work she completed:

Step 1: 3t+3−4t≥−5
Step 2: −t+3≥−5
Step 3: −t≥−2
Step 4: t≤2


(12 points)
True False
Izzie made a mistake going from step 2 to step 3

Izzie should not have changed the inequality symbol from ≥
to ≤

Izzie made a mistake going from step 1 to step 2

Izzie made a mistake going from step 3 to step 4

Izzie made no mistakes in solving this inequality

The correct answer should have been t≤8

Question 16
15. Which graph represents the solutions to this inequality: 5n−3<3(n+1)
(2 points)
Responses

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1 answer

Let's go through each question step by step and provide the correct responses:

Question 1

Which equation has the same solution for x as this equation: x - 12 = 40? To solve for x from the given equation, we can rewrite it as: \[ x = 40 + 12 \] \[ x = 52 \]

We can check which responses will yield the same solution:

  • x/4 = 13 → x = 52 (This is correct)
  • x + 12 = -40 → x = -52 (Incorrect)
  • x/2 = 14 → x = 28 (Incorrect)
  • 12 - x = 40 → x = -28 (Incorrect)

Correct response: x/4 = 13

Question 2

Length of the rectangle expression: Given that the length (L) is four meters less than twice its width (W): \[ L = 2W - 4 \]

The formula for the area of a rectangle is: \[ A = L \times W \] So, \[ 96 = (2W - 4) \times W \]

Using trial and error:

  1. Test values for W:
    • If W = 8: L = 2(8) - 4 = 12 → Area = 8 * 12 = 96 (Correct)

Responses:

  • Expression for length in terms of width: 2W - 4
  • Formula for the area of a rectangle: Area = L × W
  • Length and width: Length = 12 m, Width = 8 m

Question 3

Match the equations with their solutions:

  1. 3i + 5 = 3i + 7: No solution (since 5 cannot equal 7).
  2. 3i + 2i - 7 = 18: Solution is i = 4 (or 3i - 2i = 18 + 7).
  3. 3i + 5 = 2i - 7: Solution is i = -12.
  4. 3(2i + 7) = 6i + 21: True for all i (identity).
  5. 3(2i − 7) = 15: Solution is i = 4.

Match:

  • 3i + 5 = 3i + 7 → No Solution
  • 3i + 2i - 7 = 18 → i = 4
  • 3i + 5 = 2i - 7 → i = -12
  • 3(2i + 7) = 6i + 21 → All i
  • 3(2i − 7) = 15 → i = 4

Question 4

Solve the equation and justify each step:

  1. Step 1: Distributive Property → \(2(x + 8) = 2x + 16\)
  2. Step 2: Combine Like Terms to get \(2x + 16 = 2x + 8\)

For this equation, there are:

  • No solutions (because the simplified form leads to a contradiction).

Question 5

Match the description of the one variable equation with the number of solutions

  1. x - 7 = 7 - x: One solution (x = 7).
  2. 3(x + 5) = 3x + 5: Infinite solutions (identity).
  3. 10 − x = 25: One solution (x = -15).
  4. 2(x + 4) = 2x + 5 + 3: No solution (leads to a contradiction).

Question 6

Inequality for student spending: The total amount spent can be expressed as: \[ 24 + 2n \leq 37 \]

Correct response: 37 ≥ 2n + 24

Question 7

Solve for b in the equation A = 12(a + b): To isolate b, rearranging gives: \[ b = \frac{A}{12} - a \]

Correct response: None of the provided responses are correct, the right form should have been \( b = \frac{A}{12} - a \).

Question 8

Solve the inequality: \(-3x + 1 ≤ -47\)

  1. Subtract 1 from both sides: \[ -3x ≤ -48 \]
  2. Divide by -3 (remember to flip the inequality): \[ x ≥ 16 \]

Question 9

Student's mistake on inequality graph for -4<x: Correct: The student did not make a mistake; this graph represents the proper solution to \(x > -4\).

Question 10

Property that justifies the step from step 3 to step 4:

  • Addition Property of Equality

Question 11

Identifying mistake in solving equation:

  1. Property used in Step 1: Distributive Property.
  2. Mistake in Step 3: Correct should have been 3x=15.
  3. Correct answer for x should be x=5.

Question 12

Grandma's garden weed problem:

  • Equation: \(250 - 5t = 30\)
  • Solution: \(t = 44\) minutes.
  • Meaning: It takes 44 minutes for her to get down to 30 weeds remaining.

Question 13

Solve the formula F=9/5C+32 for C:

  1. Subtract 32 from both sides: \(F - 32 = \frac{9}{5}C\).
  2. Multiply both sides by \(\frac{5}{9}\) to solve for C.

Correct option: \(C = \frac{5}{9}(F - 32)\).

With these responses, you should have the answers to all questions. If any specific question needs more detailed expansion or clarification, let me know!