Rearrange the formula X = mb using m as the quantity of interest

To rearrange the formula \( X = mb \) to express \( m \) as the quantity of interest, we need to isolate \( m \) on one side of the equation. Here are the steps to do that:

1. Start with the original equation:
\[
X = mb
\]

2. To isolate \( m \), we need to get rid of \( b \) on the right side. We can do this by dividing both sides of the equation by \( b \):
\[
\frac{X}{b} = \frac{mb}{b}
\]

3. Simplifying the right side, the \( b \)s cancel out:
\[
\frac{X}{b} = m
\]

4. Finally, we write it in the standard form, which is:
\[
m = \frac{X}{b}
\]

So, the rearranged formula with \( m \) as the quantity of interest is:
\[
m = \frac{X}{b}
\]

That's it! Now we have \( m \) on one side by itself!

Rearrange the formula A = 3 + xy using x as the quantity of interest.

(Show your work, original, and like a 9th grader wrote it)

To rearrange the formula \( A = 3 + xy \) and make \( x \) the quantity of interest, we need to isolate \( x \) on one side of the equation. Here’s how we can do that step by step:

1. Start with the original equation:
\[
A = 3 + xy
\]

2. The first step is to get rid of the 3 on the right side. We can do this by subtracting 3 from both sides:
\[
A - 3 = xy
\]

3. Now we have \( xy \) on the right side. Next, we need to isolate \( x \). To do this, we can divide both sides by \( y \):
\[
\frac{A - 3}{y} = \frac{xy}{y}
\]

4. On the right side, the \( y \) cancels out:
\[
\frac{A - 3}{y} = x
\]

5. Finally, we can write it in the standard form as:
\[
x = \frac{A - 3}{y}
\]

So, the rearranged formula with \( x \) as the quantity of interest is:
\[
x = \frac{A - 3}{y}
\]

And that's how we isolated \( x \)!

A group of marine biologists collected data on the velocity of a dolphin using a GPS tracker. The GPS tracker measured that the dolphin was swimming at 25 knots. The biologists are interested in finding out how many hours it would take for the dolphin to swim 45 nautical miles if it continues to swim at a constant speed of 25 knots. In order to use units to verify your rearrangement of the formula, what process needs to be done? Enter the value that corresponds to your answer.

Option #1: miles
Option #2: nautical miles
Option #3: hours

I would need to convert the units of distance from knots to Option #___

(Show your work, original, and like a 9th grader wrote it)

To find out how long it would take for the dolphin to swim 45 nautical miles at a speed of 25 knots, we can use the formula:

\[
\text{time} = \frac{\text{distance}}{\text{speed}}
\]

In this case:
- Distance = 45 nautical miles
- Speed = 25 knots

Now, we need to make sure the units are consistent. The speed of 25 knots is already in nautical miles per hour, since 1 knot = 1 nautical mile per hour. This means we don't need to change the units of speed, but we can see if the distance needs to be in the same unit.

### Calculation:
1. We insert the values into the time formula:
\[
\text{time} = \frac{45 \text{ nautical miles}}{25 \text{ knots}}
\]

2. Since knots means nautical miles per hour, we can directly perform the calculation:
\[
\text{time} = \frac{45}{25} = 1.8 \text{ hours}
\]

### Conclusion:
Since knots is expressed in nautical miles per hour and we have the distance in nautical miles, we can see that we don’t need to convert anything. So, we have verified that all units make sense.

Now, to fill in the blank regarding what needs to be converted, since the distance measured in nautical miles matches the speed measured in knots, we don’t really need to convert the units for either distance or speed—they are compatible.

Therefore, the answer would be:
\[
\text{I would need to convert the units of distance from knots to Option #2: nautical miles.}
\]

So, the final statement would be:
**Option #2: nautical miles**.

There’s no need to convert, but if it was required in a different scenario, it should be in nautical miles, which is already what we have!

A group of ecologists collected data on the velocity of a river otter using a motion sensor. The motion sensor measured that the river otter was swimming at 3 meters per second. The ecologists are interested in finding out how many seconds it would take for the otter to swim 150 feet if it continues to swim at a constant speed of 3 meters per second. In order to use units to verify your rearrangement of the formula, what process needs to be done? Enter the value that corresponds to your answer.

Option #1: feet
Option #2: seconds
Option #3: meters

I would need to convert the units of distance from meters to Option #_____
(Show your work, original, and like a 9th grader wrote it)

To find out how long it will take for the river otter to swim 150 feet at a speed of 3 meters per second, we can use the formula:

\[
\text{time} = \frac{\text{distance}}{\text{speed}}
\]

### Given Data:
- Speed = 3 meters per second
- Distance = 150 feet

### Step 1: Identify Units
Before we can use the formula, we need to make sure the units are consistent. The speed is given in meters per second, and the distance is given in feet. To use the formula correctly, both units must match.

### Step 2: Convert Distance
1. We need to convert the distance from feet to meters because the speed is in meters. We know:
- 1 meter ≈ 3.28084 feet

2. To convert feet to meters, we can use the conversion factor:
\[
\text{distance in meters} = \frac{\text{distance in feet}}{3.28084}
\]

3. Now, let’s convert 150 feet to meters:
\[
\text{distance in meters} = \frac{150 \text{ feet}}{3.28084} \approx 45.72 \text{ meters}
\]

### Step 3: Use the Formula
Now we can use the formula with the converted distance:
\[
\text{time} = \frac{45.72 \text{ meters}}{3 \text{ meters/second}} = 15.24 \text{ seconds}
\]

### Conclusion
Since we converted the distance from feet to meters to match the speed, we can safely say that:

I would need to convert the units of distance from meters to **Option #3: meters** (because we actually converted feet to meters to match them with the speed).

So the final statement is:
**Option #3: meters.**

Jason is helping his friend build a rectangular garden. The area of the garden is 36 square meters. The length of the garden is 9 meters. Jason wonders what the width of the garden will be. Jason knows that the formula for the area of a rectangle is (A = l * w). How would he rearrange this formula to highlight the width, w, as the quantity of interest?

(Show your work, original, and like a 9th grader wrote it)

To find the width \( w \) of the rectangular garden using the formula for the area of a rectangle, \( A = l \times w \), Jason can follow these steps to rearrange the formula so that \( w \) is by itself:

### Step 1: Start with the Original Formula
The original formula for the area of a rectangle is:
\[
A = l \times w
\]

### Step 2: Substitute the Known Values
We know that the area \( A \) is 36 square meters and the length \( l \) is 9 meters. We can plug the values into the equation:
\[
36 = 9 \times w
\]

### Step 3: Isolate the Width \( w \)
To find \( w \), we need to isolate it on one side of the equation. We can do this by dividing both sides of the equation by the length \( l \) (which is 9 meters):
\[
w = \frac{36}{9}
\]

### Step 4: Do the Calculation
Now we just need to calculate the value of \( w \):
\[
w = 4
\]

### Conclusion
So the width \( w \) of the garden will be:
\[
w = 4 \text{ meters}
\]

In rearranging the formula, Jason first started with \( A = l \times w \) and then isolated \( w \) by dividing both sides by \( l \). Hence, the rearranged formula for the width is:
\[
w = \frac{A}{l}
\]
In this case:
\[
w = \frac{36}{9} = 4 \text{ meters}
\]

That's how Jason can find the width of the garden!

Bao is designing a rectangular swimming pool in her backyard. The area of the swimming pool will be 120 square feet. The length of the swimming pool will be 15 feet. Bao wonders what the width of the swimming pool will be. Bao knows that the formula for the area of a rectangle (A = l * w). How would she rearrange this formula to highlight the width, w, as the quantity of interest? Please also help Bao solve the problem and show your work.

(Show your work, original, and like a 9th grader wrote it)

To help Bao find the width \( w \) of the rectangular swimming pool using the area formula \( A = l \times w \), we can follow these steps to rearrange the formula and then solve for the width.

### Step 1: Start with the Original Formula
The formula for the area of a rectangle is:
\[
A = l \times w
\]

### Step 2: Substitute the Known Values
We know:
- Area \( A = 120 \) square feet
- Length \( l = 15 \) feet

Now we can plug these values into the formula:
\[
120 = 15 \times w
\]

### Step 3: Rearrange the Formula to Highlight \( w \)
To get \( w \) by itself, we need to divide both sides of the equation by the length \( l \) (which is 15 feet):
\[
w = \frac{120}{15}
\]

### Step 4: Calculate the Width
Now we just need to do the calculation:
\[
w = \frac{120}{15} = 8
\]

### Conclusion
So the width \( w \) of the swimming pool will be:
\[
w = 8 \text{ feet}
\]

### Summary of Steps
1. Start with the area formula: \( A = l \times w \).
2. Substitute the known values: \( 120 = 15 \times w \).
3. Rearrange to solve for \( w \): \( w = \frac{120}{15} \).
4. Calculate \( w \): \( w = 8 \) feet.

That's how Bao can find the width of the swimming pool!

Which set notation correctly expresses the solution set to the equation

5x + 7 = 22?
a. (x=3)
b. {3}
c. (3)
d. {x=3}

To solve the equation \(5x + 7 = 22\), we can follow these steps:

1. Subtract 7 from both sides:
\[
5x = 22 - 7
\]
\[
5x = 15
\]

2. Divide both sides by 5:
\[
x = \frac{15}{5} = 3
\]

The solution to the equation is \(x = 3\).

Now, let’s look at the options given for the correct set notation to express this solution.

The correct notation should specifically denote the solution (the value of \(x\)). The following options are provided:

- a. \((x=3)\)
- b. \(\{3\}\)
- c. \((3)\)
- d. \(\{x=3\}\)

### Analysis of the Options:
- **a. \((x=3)\)**: This notation is not a standard way to express a solution set.

- **b. \(\{3\}\)**: This option correctly expresses the solution set as a set containing only the number 3.

- **c. \((3)\)**: This notation also does not properly represent a solution set.

- **d. \(\{x=3\}\)**: This notation is not standard for expressing a solution set either.

### Conclusion:
The correct answer is:
\[
\text{b. } \{3\}
\]

This option indicates that the solution set contains the single element 3.

Which set notation correctly expresses the solution set to the equation

5x + 7 = 22?
a. (x=3)
b. {3}
c. (3)
d. {x=3}

(Show your work, original, and like a 9th grader wrote it)

To solve the equation \(5x + 7 = 22\) and express the solution in the correct set notation, let's go through the steps carefully:

### Step 1: Solve the Equation
We start with the equation:
\[
5x + 7 = 22
\]

**Step 2:** Subtract 7 from both sides to isolate the term with \(x\):
\[
5x + 7 - 7 = 22 - 7
\]
This simplifies to:
\[
5x = 15
\]

**Step 3:** Now divide both sides by 5 to solve for \(x\):
\[
x = \frac{15}{5} = 3
\]

So, the solution to the equation is:
\[
x = 3
\]

### Step 2: Express the Solution in Set Notation
Now we want to express the solution \(x = 3\) in set notation. The correct ways to express a solution set typically involve using curly braces \(\{\}\).

Let's review the given options:
- **a. \((x=3)\)**: This notation does not correctly represent a set.
- **b. \(\{3\}\)**: This notation is correct. It indicates that 3 is a member of the set (the solution set consists of the number 3).
- **c. \((3)\)**: This notation is not a proper way to express a set.
- **d. \(\{x=3\}\)**: This notation is not standard for expressing a solution set either; it suggests a different meaning.

### Conclusion
The correct way to express the solution set to the equation \(5x + 7 = 22\) is:
\[
\text{b. } \{3\}
\]

This option shows that the solution set contains one element, which is 3.

Which set notation correctly expresses the solution set to the equation 2x – 8 = 10?

a. (x=9)
b. {9}
c. (9)
d. {x=9}

(Show your work, original, and like a 9th grader wrote it)

To solve the equation \(2x - 8 = 10\) and express the solution in the correct set notation, let's go through the steps in a clear way:

### Step 1: Solve the Equation
We start with the equation:
\[
2x - 8 = 10
\]

**Step 2:** Add 8 to both sides to isolate the term with \(x\):
\[
2x - 8 + 8 = 10 + 8
\]
This simplifies to:
\[
2x = 18
\]

**Step 3:** Now, divide both sides by 2 to solve for \(x\):
\[
x = \frac{18}{2} = 9
\]

So, the solution to the equation is:
\[
x = 9
\]

### Step 2: Express the Solution in Set Notation
Now we want to express the solution \(x = 9\) in set notation. The correct way to express a solution set uses curly braces \(\{\}\).

Let's review the given options:
- **a. \((x=9)\)**: This notation does not correctly represent a set.
- **b. \(\{9\}\)**: This notation is correct. It indicates that 9 is a member of the set (the solution set consists of the number 9).
- **c. \((9)\)**: This notation is not a proper way to express a set.
- **d. \(\{x=9\}\)**: This notation is not standard for expressing a solution set either; it suggests something different.

### Conclusion
The correct way to express the solution set to the equation \(2x - 8 = 10\) is:
\[
\text{b. } \{9\}
\]

This option shows that the solution set contains one element, which is 9.

Maya, Liam, and Zoe are discussing the solution set to the equation

x^2 - 4 = 0
Maya says the solution is {2}
Liam says the solution is {-2, 2}
Zoe says the solution is {-2}
Which person has found the correct solution set?
a. Liam
b. None of them are correct
c. Zoe
d. Maya

To determine which person has found the correct solution set for the equation \(x^2 - 4 = 0\), let's first solve the equation.

### Step 1: Solve the Equation
The equation is:
\[
x^2 - 4 = 0
\]

We can factor this equation:
\[
(x - 2)(x + 2) = 0
\]

### Step 2: Apply the Zero Product Property
According to the zero product property, if the product of two factors is zero, then at least one of the factors must be zero. This gives us two equations to solve:
1. \(x - 2 = 0\)
2. \(x + 2 = 0\)

### Step 3: Solve Each Equation
1. From \(x - 2 = 0\):
\[
x = 2
\]

2. From \(x + 2 = 0\):
\[
x = -2
\]

### Step 4: Compile the Solutions
Thus, the solutions to the equation \(x^2 - 4 = 0\) are:
\[
x = 2 \quad \text{and} \quad x = -2
\]

### Step 5: Identify the Correct Solution Set
The solution set can be written as:
\[
\{-2, 2\}
\]

### Analysis of Responses
- **Maya says the solution is \{2\}**: This is not correct because it ignores the solution \(-2\).
- **Liam says the solution is \{-2, 2\}**: This is correct because it includes both valid solutions.
- **Zoe says the solution is \{-2\}**: This is incorrect because it ignores the solution \(2\).

### Conclusion
The person who found the correct solution set is:
\[
\text{a. Liam}
\]

Dante, Ahmed and Olivia are discussing the solution set to the equation

x^2 + 16 = 25
Dante says the solution is {3}
Ahmed says the solution is {-3, 3}
Olivia says the solution is {-3}
Which person has found the correct solution set?
Dante
Ahmed
Olivia
None of them are correct
(Show your work, original, and like a 9th grader wrote it)

To determine which person has found the correct solution set for the equation \(x^2 + 16 = 25\), let's solve the equation step by step.

### Step 1: Write the Original Equation
The equation we start with is:
\[
x^2 + 16 = 25
\]

### Step 2: Move 16 to the Other Side
To isolate the \(x^2\) term, we can subtract 16 from both sides:
\[
x^2 + 16 - 16 = 25 - 16
\]
This simplifies to:
\[
x^2 = 9
\]

### Step 3: Take the Square Root of Both Sides
To solve for \(x\), we take the square root of both sides. Remember, when you take the square root, you get both the positive and negative roots:
\[
x = \pm \sqrt{9}
\]
This means:
\[
x = 3 \quad \text{or} \quad x = -3
\]

### Step 4: Compile the Solutions
The solution set to the equation \(x^2 + 16 = 25\) is:
\[
\{-3, 3\}
\]

### Analysis of Responses
- **Dante says the solution is \{3\}**: This is incorrect because it only includes the positive solution.
- **Ahmed says the solution is \{-3, 3\}**: This is correct because it includes both valid solutions.
- **Olivia says the solution is \{-3\}**: This is incorrect because it only includes the negative solution.

### Conclusion
The person who found the correct solution set is:
\[
\text{Ahmed}
\]

What value falls in the solution set of the inequality

5 – 2x > 7?
a. (-1)
b. (1)
c. (0)
d. (-2)

To find which value falls in the solution set of the inequality \(5 - 2x > 7\), we first need to solve the inequality step by step.

### Step 1: Solve the Inequality
Start with the inequality:
\[
5 - 2x > 7
\]

**Step 2:** Subtract 5 from both sides:
\[
5 - 5 - 2x > 7 - 5
\]
This simplifies to:
\[
-2x > 2
\]

**Step 3:** Divide both sides by \(-2\). Remember that when we divide or multiply both sides of an inequality by a negative number, we must flip the inequality sign:
\[
x < -1
\]

### Step 2: Identify the Solution Set
The solution to the inequality is:
\[
x < -1
\]
This means any value less than \(-1\) will satisfy the inequality.

### Step 3: Check the Given Options
Now we will check the given values:
- **a. \(-1\)**: This value is not less than \(-1\), so it does not satisfy the inequality.
- **b. \(1\)**: This value is greater than \(-1\), so it does not satisfy the inequality.
- **c. \(0\)**: This value is greater than \(-1\), so it does not satisfy the inequality.
- **d. \(-2\)**: This value is less than \(-1\), so it does satisfy the inequality.

### Conclusion
The value that falls in the solution set of the inequality \(5 - 2x > 7\) is:
\[
\text{d. } (-2)
\]

The solution set of the inequality

( 2(a + 1) - 3a < 4) is the set of values of a less than 4. Which set notation correctly expresses the solutions?
a. {a : a < 4}
b. {a : a > 4}
c. {a : a < 6}
d. {a : a > 6}
(Show your work, original, and like a 9th grader wrote it)

To determine which set notation correctly expresses the solution set of the inequality \(2(a + 1) - 3a < 4\), let's first go through the steps to solve the inequality.

### Step 1: Simplify the Inequality
Start with the given inequality:
\[
2(a + 1) - 3a < 4
\]

**Step 2:** Distribute \(2\) to the terms in the parentheses:
\[
2a + 2 - 3a < 4
\]

**Step 3:** Combine like terms on the left side:
\[
(2a - 3a) + 2 < 4
\]
This simplifies to:
\[
-a + 2 < 4
\]

### Step 2: Isolate the Variable
**Step 4:** Subtract \(2\) from both sides:
\[
-a < 4 - 2
\]
This simplifies to:
\[
-a < 2
\]

**Step 5:** Multiply both sides by \(-1\). Remember that when we multiply an inequality by a negative number, we must flip the inequality sign:
\[
a > -2
\]

### Step 3: Summarize the Solution Set
So the solution we found is:
\[
a > -2
\]

However, the problem states that the solution set is the set of values of \(a\) less than 4. Based on the initial calculations, it seems like there's a discrepancy. The inequality \(2(a + 1) - 3a < 4\) leads to \(a > -2\), which means the values of \(a\) we found are greater than \(-2\).

### Step 4: Analyzing the Set Notation Options
Now let's analyze the given answer options:
- **a. \{a : a < 4\}**: This means \(a\) is less than \(4\), which is not the solution we found (\(a > -2\)).
- **b. \{a : a > 4\}**: This means \(a\) is greater than \(4\), which is not correct.
- **c. \{a : a < 6\}**: This means \(a\) is less than \(6\), which is also not directly correct.
- **d. \{a : a > 6\}**: This means \(a\) is greater than \(6\), which is not correct.

### Conclusion
The solution set of \(a > -2\) is not among the provided options, and it looks like there's a mix-up in the initial problem statement regarding the condition given.

However, based on the solution we derived (\(a > -2\)), it was found correctly, but we can conclude that:

**None of the provided options correctly express the solution. If it were specific to the inequality found, the correct representation based on the calculations would be \{a : a > -2\}.** Since that's not an available option, the conclusion is that **none of them are correct** based on the derived inequality.

try again, one os those is the correct answer

Apologies for the confusion. Let's go through the original inequality step by step again to ensure we reach the correct solution.

### Step 1: Solve the Inequality
Given:
\[
2(a + 1) - 3a < 4
\]

**Step 2:** Distribute \(2\):
\[
2a + 2 - 3a < 4
\]

**Step 3:** Combine like terms:
\[
(2a - 3a) + 2 < 4
\]
This simplifies to:
\[
-a + 2 < 4
\]

**Step 4:** Subtract \(2\) from both sides:
\[
-a < 4 - 2
\]
This simplifies to:
\[
-a < 2
\]

**Step 5:** Multiply both sides by \(-1\) (remember to flip the inequality sign):
\[
a > -2
\]

### Analyzing the Solution Set
The solution \(a > -2\) means that \(a\) can take any value greater than \(-2\). However, the problem mentioned that the solution set is values of \(a\) less than \(4\).

### Reviewing the Options
Now let's review the provided answer options again in the context of the solution \(a < 4\):

- **a. \{a : a < 4\}**: This means \(a\) is less than \(4\).

- **b. \{a : a > 4\}**: This means \(a\) is greater than \(4\).

- **c. \{a : a < 6\}**: This means \(a\) is less than \(6\).

- **d. \{a : a > 6\}**: This means \(a\) is greater than \(6\).

### Reviewing the Correct Solution Set
Given that the solution set should express values of \(a\) less than \(4\) and considering that \(a > -2\) also includes values leading towards \(4\) (but doesn't provide an upper bound), the best matching option regarding a less-than relationship that covers more of the relevant range would be:

**The correct option is:**
\[
\text{a. } \{a : a < 4\}
\]

This option expresses the condition that the values of \(a\) can be any number less than \(4\), which is aligned with the statement of what the solution set needs to be. Thank you for your patience!

Use set notation to express the value(s) that make the equation 3y = 15 true. Enter your response using set notation.

(Show your work, original, and like a 9th grader wrote it)

To express the value(s) that make the equation \(3y = 15\) true using set notation, we first need to solve the equation step by step.

### Step 1: Start with the Original Equation
The equation we have is:
\[
3y = 15
\]

### Step 2: Solve for \(y\)
To find \(y\), we need to isolate it on one side of the equation. We can do this by dividing both sides by \(3\):
\[
y = \frac{15}{3}
\]

### Step 3: Calculate the Value of \(y\)
Now we perform the division:
\[
y = 5
\]

### Step 4: Express the Solution in Set Notation
The solution to the equation \(3y = 15\) gives us \(y = 5\). In set notation, we write this as:
\[
\{5\}
\]

### Conclusion
So, the value that makes the equation \(3y = 15\) true, expressed in set notation, is:
\[
\{5\}
\]

This notation indicates that the solution set contains only the single value \(5\).

Use set notation to express the value(s) that make the equation 4z + 8 = 0 true. Enter your response using set notation.

(Show your work, original, and like a 9th grader wrote it)

To express the value(s) that make the equation \(4z + 8 = 0\) true using set notation, let's go through the steps to solve the equation.

### Step 1: Start with the Original Equation
The equation we have is:
\[
4z + 8 = 0
\]

### Step 2: Isolate the Variable
To isolate \(z\), we first need to get rid of the \(8\) on the left side. We can do this by subtracting \(8\) from both sides:
\[
4z + 8 - 8 = 0 - 8
\]
This simplifies to:
\[
4z = -8
\]

### Step 3: Solve for \(z\)
Now we will divide both sides by \(4\) to solve for \(z\):
\[
z = \frac{-8}{4}
\]

### Step 4: Calculate the Value
Performing the division gives us:
\[
z = -2
\]

### Step 5: Express the Solution in Set Notation
The solution to the equation \(4z + 8 = 0\) gives us \(z = -2\). In set notation, we can write this as:
\[
\{-2\}
\]

### Conclusion
So, the value that makes the equation \(4z + 8 = 0\) true, expressed in set notation, is:
\[
\{-2\}
\]

This indicates that the solution set contains only the single value \(-2\).

The solution to the inequality 3x - 2 < 7 is x < 3. Use set notation to express the values of x that make the inequality 3x - 2 < 7 true.

(Show your work, original, and like a 9th grader wrote it)

To express the values of \(x\) that make the inequality \(3x - 2 < 7\) true using set notation, let’s first go through the steps to solve the inequality.

### Step 1: Start with the Original Inequality
The inequality we have is:
\[
3x - 2 < 7
\]

### Step 2: Isolate the Variable
To isolate \(x\), we start by adding \(2\) to both sides of the inequality:
\[
3x - 2 + 2 < 7 + 2
\]
This simplifies to:
\[
3x < 9
\]

### Step 3: Solve for \(x\)
Next, we divide both sides by \(3\):
\[
x < \frac{9}{3}
\]
This simplifies to:
\[
x < 3
\]

### Step 4: Express the Solution in Set Notation
The solution \(x < 3\) means that \(x\) can take any value less than \(3\). In set notation, we express this as:
\[
\{ x \mid x < 3 \}
\]

### Conclusion
So, the values of \(x\) that make the inequality \(3x - 2 < 7\) true, expressed in set notation, is:
\[
\{ x \mid x < 3 \}
\]

This indicates the set of all \(x\) that are less than \(3\).

The solution to the inequality 6y - 4 > 8 is y > 2. Use set notation to express the values of y that make the inequality 6y - 4 > 8 true.

(Show your work, original, and like a 9th grader wrote it)

To express the values of \(y\) that make the inequality \(6y - 4 > 8\) true using set notation, let's go through the steps of solving the inequality.

### Step 1: Start with the Original Inequality
The inequality we have is:
\[
6y - 4 > 8
\]

### Step 2: Isolate the Variable
To isolate \(y\), we start by adding \(4\) to both sides of the inequality:
\[
6y - 4 + 4 > 8 + 4
\]
This simplifies to:
\[
6y > 12
\]

### Step 3: Solve for \(y\)
Next, we divide both sides by \(6\):
\[
y > \frac{12}{6}
\]
This simplifies to:
\[
y > 2
\]

### Step 4: Express the Solution in Set Notation
The solution \(y > 2\) means that \(y\) can take any value greater than \(2\). In set notation, we express this as:
\[
\{ y \mid y > 2 \}
\]

### Conclusion
So, the values of \(y\) that make the inequality \(6y - 4 > 8\) true, expressed in set notation, is:
\[
\{ y \mid y > 2 \}
\]

This notation indicates the set of all \(y\) that are greater than \(2\).

Determine whether the following linear equations have the same solution set.

4x + 10 = 34
2(x + 5) + 12 = 38
(Show your work, original, and like a 9th grader wrote it)

To determine whether the linear equations \(4x + 10 = 34\) and \(2(x + 5) + 12 = 38\) have the same solution set, let's solve each equation step by step.

### Step 1: Solve the First Equation
The first equation is:
\[
4x + 10 = 34
\]

**Step 1.1:** Subtract \(10\) from both sides:
\[
4x + 10 - 10 = 34 - 10
\]
This simplifies to:
\[
4x = 24
\]

**Step 1.2:** Divide both sides by \(4\) to solve for \(x\):
\[
x = \frac{24}{4}
\]
So:
\[
x = 6
\]

### Step 2: Solve the Second Equation
The second equation is:
\[
2(x + 5) + 12 = 38
\]

**Step 2.1:** First, simplify \(2(x + 5)\):
\[
2x + 10 + 12 = 38
\]
This can be simplified further:
\[
2x + 22 = 38
\]

**Step 2.2:** Subtract \(22\) from both sides:
\[
2x + 22 - 22 = 38 - 22
\]
This simplifies to:
\[
2x = 16
\]

**Step 2.3:** Divide both sides by \(2\) to solve for \(x\):
\[
x = \frac{16}{2}
\]
So:
\[
x = 8
\]

### Step 3: Compare the Solutions
Now we have the solutions:
- From the first equation, we found \(x = 6\).
- From the second equation, we found \(x = 8\).

### Conclusion
Since the solutions \(6\) and \(8\) are not the same, we can conclude that the two linear equations do not have the same solution set.

Thus, the final answer is:
**No, the equations \(4x + 10 = 34\) and \(2(x + 5) + 12 = 38\) do not have the same solution set.**

What value of q will make the following equation true?

1/5 (10 - 5q) - 3 = q
(Show your work, original, and like a 9th grader wrote it)

To find the value of \(q\) that makes the equation \(\frac{1}{5}(10 - 5q) - 3 = q\) true, let's solve it step by step.

### Step 1: Start with the Original Equation
The equation we have is:
\[
\frac{1}{5}(10 - 5q) - 3 = q
\]

### Step 2: Eliminate the Fraction
To make it easier to work with, we can eliminate the fraction. We will first distribute the \(\frac{1}{5}\):
\[
\frac{1}{5} \times 10 - \frac{1}{5} \times 5q - 3 = q
\]
This simplifies to:
\[
2 - q - 3 = q
\]

### Step 3: Combine Like Terms
Now, let's combine the terms on the left side:
\[
2 - 3 - q = q
\]
This simplifies to:
\[
-1 - q = q
\]

### Step 4: Move Terms to One Side
Now we want to isolate \(q\). To do this, we can add \(q\) to both sides:
\[
-1 = q + q
\]
This simplifies to:
\[
-1 = 2q
\]

### Step 5: Solve for \(q\)
Now, we divide both sides by \(2\) to solve for \(q\):
\[
q = \frac{-1}{2}
\]

### Conclusion
The value of \(q\) that makes the equation \(\frac{1}{5}(10 - 5q) - 3 = q\) true is:
\[
q = -\frac{1}{2}
\]

So, the final answer is:
\[
q = -0.5 \text{ or } q = -\frac{1}{2}
\]

For the last question, did you start first with the distributive property or the multiplication property of equality? Explain your selection. What would have happened if you started with the other strategy?

(Show your work, original, and like a 9th grader wrote it)

In the last question, I started with the **distributive property** by distributing \(\frac{1}{5}\) to both \(10\) and \(-5q\) in the equation \(\frac{1}{5}(10 - 5q) - 3 = q\). Here’s a breakdown of my reason for selecting that approach and what would have happened if I had used the other strategy.

### Reason for Using the Distributive Property First
1. **Clarity**: By distributing \(\frac{1}{5}\), I converted the left side of the equation into a more manageable form by directly simplifying \(\frac{1}{5}(10 - 5q)\).

2. **Simplicity**: It helped to express the equation without any fractions, which often makes it easier to handle. Fractions can complicate equations, so eliminating them early on can streamline the solving process.

### What If I Had Used the Multiplication Property of Equality First?
If I had used the **multiplication property of equality** first, I would have multiplied both sides of the equation by \(5\) to eliminate the fraction:
\[
\frac{1}{5}(10 - 5q) - 3 = q
\]
Multiplying all terms in the equation by \(5\):
\[
10 - 5q - 15 = 5q
\]
This gives us:
\[
-5q - 5 = 5q
\]

### Continuing with That Approach:
1. **Combine Like Terms**:
\[
-5 - 5 = 5q + 5q
\]
which simplifies to:
\[
-5 = 10q
\]

2. **Solve for \(q\)**:
\[
q = \frac{-5}{10} = -\frac{1}{2}
\]

**Conclusion**:
Regardless of whether I used the **distributive property** first or the **multiplication property of equality**, I would arrive at the same solution for \(q = -\frac{1}{2}\). However, using the distributive property first allowed for a more straightforward approach in this case, keeping the equation clean from the start. Each method would ultimately yield the same answer because they are both valid algebraic techniques, but the distributive approach reduced complexity in the early steps.

Darnell was asked to use and identify appropriate properties of equality to find the value of d that makes this statement -4d – 8 = 2 + d true. He has written out his steps below. Label those steps with the property used.

-4d + 4d – 8 = 2 + d + 4d (________________)
-8 = 2 + 5d (combine like terms)
-8 – 2 = 2 – 2 + 5d (_________________)
-10 = 5d (combine like terms)
-10/5 = 5d/5 (______________________)
-2 = d
(Show your work, original, and like a 9th grader wrote it)

Let's identify the properties of equality Darnell used to solve the equation \(-4d - 8 = 2 + d\) and label those steps.

### Step 1: Start with the Original Equation
The equation is:
\[
-4d - 8 = 2 + d
\]

### Darnell's Steps with Properties

1. **First Step:**
\[
-4d + 4d - 8 = 2 + d + 4d
\]
This step eliminates the \(-d\) from the right side by adding \(4d\) to both sides. This process uses the **Addition Property of Equality**, which states if you add the same amount to both sides of an equation, the equation remains balanced.
- Property: **Addition Property of Equality**

2. **Second Step:**
\[
-8 = 2 + 5d
\]
In this step, Darnell combined like terms on both sides of the equation. This doesn't have a formal name but is simply referred to as **combining like terms**.

3. **Third Step:**
\[
-8 - 2 = 2 - 2 + 5d
\]
In this step, he subtracts \(2\) from both sides. This process uses the **Subtraction Property of Equality**, which states if you subtract the same amount from both sides of an equation, the equation remains balanced.
- Property: **Subtraction Property of Equality**

4. **Fourth Step:**
\[
-10 = 5d
\]
Again, he combined like terms here. This step is another instance of **combining like terms**.

5. **Fifth Step:**
\[
-\frac{10}{5} = \frac{5d}{5}
\]
Here, Darnell divided both sides by \(5\) to isolate \(d\). This step uses the **Division Property of Equality**, which states that if you divide both sides of an equation by the same non-zero number, the equation remains balanced.
- Property: **Division Property of Equality**

### Final Result
So, Darnell's work with the identified properties looks like this:

1. **Step 1:** \(-4d + 4d - 8 = 2 + d + 4d\) (Addition Property of Equality)
2. **Step 2:** \(-8 = 2 + 5d\) (combine like terms)
3. **Step 3:** \(-8 - 2 = 2 - 2 + 5d\) (Subtraction Property of Equality)
4. **Step 4:** \(-10 = 5d\) (combine like terms)
5. **Step 5:** \(-10/5 = 5d/5\) (Division Property of Equality)

### Conclusion
This way, Darnell can clearly explain each step he took to solve the equation and properly label each corresponding property of equality. The final value he derived for \(d\) is:
\[
d = -2
\]

Determine whether the following equations have the same solution set.

4y - 9 = 3y + 15
5y + 20 = 35
2b - 4b + 6b = 12
(Show your work, original, and like a 9th grader wrote it)

To determine whether the given equations have the same solution set, let's solve each of them step by step.

### Equation 1: \(4y - 9 = 3y + 15\)

**Step 1:** Start with the original equation.
\[
4y - 9 = 3y + 15
\]

**Step 2:** Subtract \(3y\) from both sides to isolate \(y\):
\[
4y - 3y - 9 = 15
\]
This simplifies to:
\[
y - 9 = 15
\]

**Step 3:** Now add \(9\) to both sides:
\[
y = 15 + 9
\]
This simplifies to:
\[
y = 24
\]

### Equation 2: \(5y + 20 = 35\)

**Step 1:** Start with the original equation.
\[
5y + 20 = 35
\]

**Step 2:** Subtract \(20\) from both sides to isolate \(y\):
\[
5y = 35 - 20
\]
This simplifies to:
\[
5y = 15
\]

**Step 3:** Now divide by \(5\) to solve for \(y\):
\[
y = \frac{15}{5}
\]
This simplifies to:
\[
y = 3
\]

### Equation 3: \(2b - 4b + 6b = 12\)

**Step 1:** Combine like terms on the left side:
\[
(2b - 4b + 6b) = 12
\]
Combine \(2b - 4b + 6b\):
\[
(2 - 4 + 6)b = 12
\]
This simplifies to:
\[
4b = 12
\]

**Step 2:** Now divide both sides by \(4\) to solve for \(b\):
\[
b = \frac{12}{4}
\]
This simplifies to:
\[
b = 3
\]

### Summary of Solutions
- From the first equation \(4y - 9 = 3y + 15\), we found:
\[
y = 24
\]
- From the second equation \(5y + 20 = 35\), we found:
\[
y = 3
\]
- From the third equation \(2b - 4b + 6b = 12\), we found:
\[
b = 3
\]

### Conclusion
Now let’s compare the solutions:
- The first equation yields \(y = 24\).
- The second equation yields \(y = 3\) and the third equation also yields \(b = 3\).

The first equation has a different solution than the other two equations. Therefore:

**The equations do not have the same solution set.**

In summary:
- Equation 1: \(y = 24\)
- Equation 2: \(y = 3\)
- Equation 3: \(b = 3\)

So, while Equations 2 and 3 both have the same solution, Equation 1 is different.

Determine whether the following equations have the same solution set.

7z + 5 = 2z + 30
3z - 10 = 5
c - 5c + 9c = 27
(Show your work, original, and like a 9th grader wrote it)

To determine whether the given equations have the same solution set, let’s solve each equation step by step.

### Equation 1: \(7z + 5 = 2z + 30\)

**Step 1:** Start with the original equation.
\[
7z + 5 = 2z + 30
\]

**Step 2:** Subtract \(2z\) from both sides to isolate the \(z\) terms:
\[
7z - 2z + 5 = 30
\]
This simplifies to:
\[
5z + 5 = 30
\]

**Step 3:** Subtract \(5\) from both sides:
\[
5z = 30 - 5
\]
This simplifies to:
\[
5z = 25
\]

**Step 4:** Now divide both sides by \(5\) to solve for \(z\):
\[
z = \frac{25}{5}
\]
This simplifies to:
\[
z = 5
\]

### Equation 2: \(3z - 10 = 5\)

**Step 1:** Start with the original equation.
\[
3z - 10 = 5
\]

**Step 2:** Add \(10\) to both sides to isolate \(z\):
\[
3z = 5 + 10
\]
This simplifies to:
\[
3z = 15
\]

**Step 3:** Now divide both sides by \(3\):
\[
z = \frac{15}{3}
\]
This simplifies to:
\[
z = 5
\]

### Equation 3: \(c - 5c + 9c = 27\)

**Step 1:** Combine like terms on the left side:
\[
(c - 5c + 9c) = 27
\]
This simplifies to:
\[
(1 - 5 + 9)c = 27
\]
Which simplifies further to:
\[
5c = 27
\]

**Step 2:** Now divide both sides by \(5\):
\[
c = \frac{27}{5}
\]
This means:
\[
c = 5.4
\]

### Summary of Solutions
- From Equation 1, we found \(z = 5\).
- From Equation 2, we found \(z = 5\).
- From Equation 3, we found \(c = 5.4\).

### Conclusion
Now let’s compare the solutions:
- Equations 1 and 2 both yield \(z = 5\).
- Equation 3 yields \(c = 5.4\).

Since Equations 1 and 2 have the same solution while Equation 3 has a different solution:

**The equations do not have the same solution set.**

In summary:
- Equation 1: \(z = 5\)
- Equation 2: \(z = 5\)
- Equation 3: \(c = 5.4\)

Thus, the first two equations have the same solution, while the last one does not match those solutions.

What value will make the equation

5n + 7/3 - 2n/3 =1 true? Write your answer as a fraction.
(Show your work, original, and like a 9th grader wrote it)

To find the value of \(n\) that makes the equation \(5n + \frac{7}{3} - \frac{2n}{3} = 1\) true, let's solve the equation step by step.

### Step 1: Start with the Original Equation
The equation we have is:
\[
5n + \frac{7}{3} - \frac{2n}{3} = 1
\]

### Step 2: Combine Like Terms
First, we need to combine the \(n\) terms. We can rewrite \(5n\) in terms of thirds (since we have fractions involving \(\frac{2n}{3}\)):
\[
5n = \frac{15n}{3}
\]
So, the equation now looks like this:
\[
\frac{15n}{3} - \frac{2n}{3} + \frac{7}{3} = 1
\]

Now combine the \(n\) terms on the left side:
\[
\frac{15n - 2n + 7}{3} = 1
\]
This simplifies to:
\[
\frac{13n + 7}{3} = 1
\]

### Step 3: Eliminate the Fraction
To eliminate the fraction, multiply both sides by \(3\):
\[
13n + 7 = 3
\]

### Step 4: Isolate \(n\)
Subtract \(7\) from both sides:
\[
13n = 3 - 7
\]
This simplifies to:
\[
13n = -4
\]

### Step 5: Solve for \(n\)
Now divide both sides by \(13\):
\[
n = \frac{-4}{13}
\]

### Conclusion
The value that makes the equation \(5n + \frac{7}{3} - \frac{2n}{3} = 1\) true is:
\[
n = \frac{-4}{13}
\]

This is the final answer expressed as a fraction.

What value will make the equation

4p - 3/5 + p/5 = 2 true? Write your answer as a fraction.
(Show your work, original, and like a 9th grader wrote it)

To find the value of \(p\) that makes the equation

\[
4p - \frac{3}{5} + \frac{p}{5} = 2
\]

true, let's solve it step by step.

### Step 1: Start with the Original Equation
The equation we have is:
\[
4p - \frac{3}{5} + \frac{p}{5} = 2
\]

### Step 2: Combine Like Terms
To combine the \(p\) terms, we need to express \(4p\) in terms of fifths since we have fractions involving \(\frac{p}{5}\).

We can rewrite \(4p\) as:
\[
4p = \frac{20p}{5}
\]

Now, substituting this into the equation gives us:
\[
\frac{20p}{5} + \frac{p}{5} - \frac{3}{5} = 2
\]

Now combine the \(p\) terms:
\[
\frac{20p + p - 3}{5} = 2
\]

This simplifies to:
\[
\frac{21p - 3}{5} = 2
\]

### Step 3: Eliminate the Fraction
To eliminate the fraction, multiply both sides by \(5\):
\[
21p - 3 = 10
\]

### Step 4: Isolate \(p\)
Add \(3\) to both sides:
\[
21p = 10 + 3
\]
This simplifies to:
\[
21p = 13
\]

### Step 5: Solve for \(p\)
Now divide both sides by \(21\):
\[
p = \frac{13}{21}
\]

### Conclusion
The value that makes the equation

\[
4p - \frac{3}{5} + \frac{p}{5} = 2
\]

true is:
\[
p = \frac{13}{21}
\]

This is the final answer expressed as a fraction.

Fatima was asked to use and identify appropriate properties of equality to find the value of y that makes the statement ¾ (y + 8) = ½ (2y + 4) true. She has written out her steps below. Label those steps with the property used.

4 (¾ (y + 8)) = 4(¼ (2y + 4)) (________________)
3 (y + 8) = 2y + 4 (simplify)
3(y) + 3(8) = 2y + 4 (____________________)
3y + 24 = 2y + 4 (simplify)
3y – 2y + 24 = 2y – 2y + 4 (_________________)
y + 24 = 4 (combine like terms)
y + 24 – 24 = 4 – 24 (_____________________)
y = -20
(Show your work, original, and like a 9th grader wrote it)

Let's go through Fatima's steps to identify the properties of equality that she used while solving the equation \(\frac{3}{4}(y + 8) = \frac{1}{2}(2y + 4)\). I'll label the steps with the appropriate properties.

### Step 1: Start with the Original Equation
Fatima starts with:
\[
\frac{3}{4}(y + 8) = \frac{1}{2}(2y + 4)
\]

### Step 2: Label the First Step
Fatima rewrites the equation as:
\[
4 \left(\frac{3}{4}(y + 8)\right) = 4\left(\frac{1}{2}(2y + 4)\right)
\]
In this step, she multiplied both sides by \(4\). This uses the **Multiplication Property of Equality**, which states that if you multiply both sides of an equation by the same number, the equation remains balanced.
- **Property:** **Multiplication Property of Equality**

### Step 3: Simplify the Equation
Fatima simplifies the equation:
\[
3(y + 8) = 2y + 4
\]
This simplification does not specifically reference a property, but it's just applying arithmetic to rewrite the equation.

### Step 4: Label the Next Step
Fatima expands the left side:
\[
3(y) + 3(8) = 2y + 4
\]
This step uses the **Distributive Property**, which states that \(a(b + c) = ab + ac\).
- **Property:** **Distributive Property**

### Step 5: Simplify Again
Fatima simplifies again to:
\[
3y + 24 = 2y + 4
\]
This is also just a simplification from the previous step.

### Step 6: Label the Next Step
Fatima rearranges the equation:
\[
3y - 2y + 24 = 2y - 2y + 4
\]
In this step, she subtracts \(2y\) from both sides. This uses the **Subtraction Property of Equality**, which states that if you subtract the same number from both sides of an equation, the equation remains balanced.
- **Property:** **Subtraction Property of Equality**

### Step 7: Combine Like Terms
Fatima now has:
\[
y + 24 = 4
\]
This step is combining like terms.

### Step 8: Label the Final Step
Now she isolates \(y\):
\[
y + 24 - 24 = 4 - 24
\]
In this step, she subtracts \(24\) from both sides. This again uses the **Subtraction Property of Equality** because she is subtracting the same quantity (\(24\)) from both sides.
- **Property:** **Subtraction Property of Equality**

### Conclusion
After labeling the steps, here’s the breakdown of properties used:

1. \(4 \left(\frac{3}{4}(y + 8)\right) = 4\left(\frac{1}{2}(2y + 4)\right)\) (Multiplication Property of Equality)
2. \(3(y + 8) = 2y + 4\) (simplify)
3. \(3(y) + 3(8) = 2y + 4\) (Distributive Property)
4. \(3y + 24 = 2y + 4\) (simplify)
5. \(3y - 2y + 24 = 2y - 2y + 4\) (Subtraction Property of Equality)
6. \(y + 24 = 4\) (combine like terms)
7. \(y + 24 - 24 = 4 - 24\) (Subtraction Property of Equality)

Fatima correctly identified how to manipulate the equation to find the value of \(y\), which concludes with:
\[
y = -20
\]

How many solutions will the equation

½ (4n + 6) – 3 = 2n have?
a. no solution
b. infinite/many solutions
c. two solutions
d. one solution

To determine how many solutions the equation

\[
\frac{1}{2}(4n + 6) - 3 = 2n
\]

has, let's solve it step by step.

### Step 1: Start with the Original Equation
The equation is:
\[
\frac{1}{2}(4n + 6) - 3 = 2n
\]

### Step 2: Distribute the \(\frac{1}{2}\)
We begin by distributing the \(\frac{1}{2}\) to the terms inside the parentheses:
\[
\frac{1}{2} \cdot 4n + \frac{1}{2} \cdot 6 - 3 = 2n
\]
This simplifies to:
\[
2n + 3 - 3 = 2n
\]

### Step 3: Combine Like Terms
Now we can combine like terms on the left side:
\[
2n + 3 - 3 = 2n
\]
Which simplifies to:
\[
2n = 2n
\]

### Step 4: Analyze the Result
The equation \(2n = 2n\) is always true for all values of \(n\). This means that every value of \(n\) satisfies the equation.

### Conclusion
Since the equation is true for all values of \(n\), it has **infinite solutions**.

Thus, the correct answer is:
\[
\text{b. infinite/many solutions}
\]

What property of inequality justifies the steps shown in the following solution process?

2/5x ≤ - 10
5/2(2/5)x ≤ - 10 (5/2)
x ≤ - 25
a. Addition Property of Inequality
b. Division Property of Inequality
c. Multiplication Property of Inequality

(Show your work, original, and like a 9th grader wrote it)
d. Subtraction Property of Inequality

To determine which property of inequality justifies the steps shown in the solution process, let’s first analyze the steps given:

1. The original inequality is:
\[
\frac{2}{5}x \leq -10
\]

2. The next step shows:
\[
\frac{5}{2}\left(\frac{2}{5}x\right) \leq -10 \left(\frac{5}{2}\right)
\]

3. The result after performing the multiplication is:
\[
x \leq -25
\]

### Analyzing the Steps
- In the second step, \(\frac{5}{2}\) is multiplied by both sides of the inequality. This implies that a number (in this case, \(\frac{5}{2}\)) is being used to multiply both sides of the inequality.

### Identifying the Correct Property
Based on the manipulation shown:

- **Multiplication Property of Inequality:** This property states that if you multiply both sides of an inequality by a positive number, the inequality still holds. In this case, multiplying both sides of the inequality by \(\frac{5}{2}\) (a positive number) is what justifies this operation.

### Conclusion
The correct answer is:
\[
\text{c. Multiplication Property of Inequality}
\]

### Explanation
This property allows us to show the steps taken in the solution process, confirming that we can manipulate the inequality this way while preserving its truth. Thus, the steps correctly reflect applying the Multiplication Property of Inequality since you multiplied both sides by a positive value (\frac{5}{2}) to isolate \(x\).

Solve the inequality

3/7 x ≥ 15
(Show your work, original, and like a 9th grader wrote it)

To solve the inequality

\[
\frac{3}{7}x \geq 15,
\]

let's go through the steps one by one.

### Step 1: Start with the Original Inequality
The inequality we have is:

\[
\frac{3}{7}x \geq 15.
\]

### Step 2: Eliminate the Fraction
To eliminate the fraction, we can multiply both sides of the inequality by the reciprocal of \(\frac{3}{7}\), which is \(\frac{7}{3}\). Remember that since we are multiplying by a positive number, the inequality sign will not change.

So, we multiply both sides by \(\frac{7}{3}\):

\[
\frac{7}{3} \cdot \frac{3}{7}x \geq 15 \cdot \frac{7}{3}.
\]

### Step 3: Simplify Both Sides
On the left side, \(\frac{7}{3} \cdot \frac{3}{7}\) simplifies to \(1\), so we are left with:

\[
x \geq 15 \cdot \frac{7}{3}.
\]

Now let's calculate \(15 \cdot \frac{7}{3}\).

### Step 4: Compute \(15 \cdot \frac{7}{3}\)
First, we can rewrite \(15\) as \(\frac{15}{1}\):

\[
15 \cdot \frac{7}{3} = \frac{15 \cdot 7}{1 \cdot 3} = \frac{105}{3}.
\]

Now, simplify \(\frac{105}{3}\):

\[
\frac{105}{3} = 35.
\]

### Step 5: Write the Final Inequality
So now we have:

\[
x \geq 35.
\]

### Conclusion
The solution to the inequality \(\frac{3}{7}x \geq 15\) is:

\[
x \geq 35.
\]

This can also be expressed in interval notation as \([35, \infty)\), which indicates that \(x\) can be any number greater than or equal to \(35\).

Solve the inequality

2/3 y ≥ 15
(Show your work, original, and like a 9th grader wrote it)

To solve the inequality

\[
\frac{2}{3}y \geq 15,
\]

let’s go through the steps clearly.

### Step 1: Start with the Original Inequality
The inequality we have is:

\[
\frac{2}{3}y \geq 15.
\]

### Step 2: Eliminate the Fraction
To get rid of the fraction, we can multiply both sides of the inequality by the reciprocal of \(\frac{2}{3}\), which is \(\frac{3}{2}\). Since we’re multiplying by a positive number, the direction of the inequality will not change.

So we multiply both sides by \(\frac{3}{2}\):

\[
\frac{3}{2} \cdot \frac{2}{3}y \geq 15 \cdot \frac{3}{2}.
\]

### Step 3: Simplify Both Sides
On the left side, \(\frac{3}{2} \cdot \frac{2}{3}\) simplifies to \(1\), which gives us:

\[
y \geq 15 \cdot \frac{3}{2}.
\]

Now we need to compute \(15 \cdot \frac{3}{2}\).

### Step 4: Compute \(15 \cdot \frac{3}{2}\)
First, we can rewrite \(15\) as \(\frac{15}{1}\):

\[
15 \cdot \frac{3}{2} = \frac{15 \cdot 3}{1 \cdot 2} = \frac{45}{2}.
\]

### Step 5: Write the Final Inequality
So now we have:

\[
y \geq \frac{45}{2}.
\]

### Step 6: Convert to Mixed Number (Optional)
If you want to express \(\frac{45}{2}\) as a mixed number:

\[
\frac{45}{2} = 22.5 \quad \text{(or \(22 \frac{1}{2}\))}.
\]

### Conclusion
The solution to the inequality \(\frac{2}{3}y \geq 15\) is:

\[
y \geq \frac{45}{2} \quad \text{(or \(y \geq 22.5\))}.
\]

### Final Result
In interval notation, this can be expressed as:

\[
\left[\frac{45}{2}, \infty\right) \quad \text{(or \([22.5, \infty)\)})
\]

indicating that \(y\) can be any number greater than or equal to \(\frac{45}{2}\) (or \(22.5\)).

Solve the inequality

5/6 x ≥ 36
(Show your work, original, and like a 9th grader wrote it)

To solve the inequality

\[
\frac{5}{6} x \geq 36,
\]

let’s break it down step by step.

### Step 1: Start with the Original Inequality
The inequality is:
\[
\frac{5}{6} x \geq 36.
\]

### Step 2: Eliminate the Fraction
To get rid of the fraction, we can multiply both sides of the inequality by the reciprocal of \(\frac{5}{6}\), which is \(\frac{6}{5}\). Since we're multiplying by a positive number, the direction of the inequality will not change.

So we multiply both sides by \(\frac{6}{5}\):

\[
\frac{6}{5} \cdot \frac{5}{6} x \geq 36 \cdot \frac{6}{5}.
\]

### Step 3: Simplify Both Sides
On the left side, \(\frac{6}{5} \cdot \frac{5}{6}\) simplifies to \(1\), giving us:

\[
x \geq 36 \cdot \frac{6}{5}.
\]

Now, we need to calculate \(36 \cdot \frac{6}{5}\).

### Step 4: Compute \(36 \cdot \frac{6}{5}\)
We can rewrite \(36\) as \(\frac{36}{1}\):

\[
36 \cdot \frac{6}{5} = \frac{36 \cdot 6}{1 \cdot 5} = \frac{216}{5}.
\]

### Step 5: Write the Final Result
So now we have:

\[
x \geq \frac{216}{5}.
\]

### Step 6: Convert to Mixed Number (Optional)
If you'd like to express \(\frac{216}{5}\) as a mixed number:

\[
\frac{216}{5} = 43.2 \quad \text{(or as a mixed number, \(43 \frac{1}{5}\))}.
\]

### Conclusion
The solution to the inequality \(\frac{5}{6} x \geq 36\) is:

\[
x \geq \frac{216}{5} \quad \text{(or \(x \geq 43.2\))}.
\]

### Final Result in Interval Notation
In interval notation, this can be expressed as:

\[
\left[\frac{216}{5}, \infty\right) \quad \text{(or \([43.2, \infty)\)})
\]

indicating that \(x\) can be any number greater than or equal to \(\frac{216}{5}\) (or \(43.2\)).