Asked by I_EAT_EVERYDAY
Solve the following:
ax − 44 < 15
ax + 15 = 44
option:
1. x = 29a
2. x = 29 over a
3. x = 59 over a
4. x < 59 over a
5. x < 29a
6. x < 59a
7. x = 59a
8. x < 29 over a
ax − 44 < 15
ax + 15 = 44
option:
1. x = 29a
2. x = 29 over a
3. x = 59 over a
4. x < 59 over a
5. x < 29a
6. x < 59a
7. x = 59a
8. x < 29 over a
Answers
Answered by
I_EAT_EVERYDAY
Solve the equation filling in the steps: 2(x−3)+21=−3
Distribute to get _____________________
Combine like terms to get ________________
Use the (choose a property) ______________ to get (resulting equation) _____________
Use the (choose a property) ______________ to get (resulting answer) _______________
Distribute to get _____________________
Combine like terms to get ________________
Use the (choose a property) ______________ to get (resulting equation) _____________
Use the (choose a property) ______________ to get (resulting answer) _______________
Answered by
I_EAT_EVERYDAY
Put the steps in order to solve the equation: 3n + 9 + 4n = 2
Step 1: ____________ to get ______
Step 2: ____________ to get ______
Step 3: ____________ to get ______
Step 1: ____________ to get ______
Step 2: ____________ to get ______
Step 3: ____________ to get ______
Answered by
I_EAT_EVERYDAY
Which equation has the same first step to solve as this equation: 2x + 12 − 3x = 27
options:
A). 4x - 17 + 2x = 9
B). 2x + 12 = −3x − 27
C). 2(x − 5) + 2x = 10
D). 2x + 12 = 27
options:
A). 4x - 17 + 2x = 9
B). 2x + 12 = −3x − 27
C). 2(x − 5) + 2x = 10
D). 2x + 12 = 27
Answered by
I_EAT_EVERYDAY
Using the Distributive Property as a good first step to solving the equation 7(5x + 2) = −4(6 − 5x), you could simplify this equation to get which of these choices?
options:
A). 35x + 14 = −24 + 20x
B). 35x + 2 = −24 − 5x
C). 5x + 14 = 6 + 20x
D). 35x + 2 = 24 − 20x
E). 35x + 14 = −24 − 20x
options:
A). 35x + 14 = −24 + 20x
B). 35x + 2 = −24 − 5x
C). 5x + 14 = 6 + 20x
D). 35x + 2 = 24 − 20x
E). 35x + 14 = −24 − 20x
Answered by
I_EAT_EVERYDAY
For the equation 2(x − 5) = 9 − 3x + 6 + 8 + 3x + 7, the left side of the equation can be simplified using the Distributive Property, and the right side of the equation can be simplified by combining like terms.
Simplify the left side of the equation:___________
Simplify the right side of the equation: ___________
Simplify the left side of the equation:___________
Simplify the right side of the equation: ___________
Answered by
I_EAT_EVERYDAY
Jill solves the equation 3(x − 5) + 2x = 0. Fill in the blanks for each step with the correct values:
3x + ____________ + 2x = 0
5x = _____________
x = __________
3x + ____________ + 2x = 0
5x = _____________
x = __________
Answered by
I_EAT_EVERYDAY
After combining like terms to simplify the equation 3 − 15x + 24 + 16x = 4x − 24 − 4x , what would be the next best step to finish solving?
A). Subtract 24 from both sides of the equation.
B). Add x to both sides of the equation.
C). Subtract 27 from both sides of the equation.
D). Divide both sides of the equation by 15.
E). Subtract x from both sides of the equation.
A). Subtract 24 from both sides of the equation.
B). Add x to both sides of the equation.
C). Subtract 27 from both sides of the equation.
D). Divide both sides of the equation by 15.
E). Subtract x from both sides of the equation.
Answered by
I_EAT_EVERYDAY
The speed limit sign on the expressway says that the speed limit is 55 mph. Write an inequality that represents all speeds you can legally drive on the expressway.
x > 55
x ≤ 55
x < 55
x ≥ 55
x > 55
x ≤ 55
x < 55
x ≥ 55
Answered by
I_EAT_EVERYDAY
If you were to graph the inequality 35 < 7x on a number line:
Would you use an open or closed circle? _________________
Would your arrow on the number line point to the left or to the right?_____
Would you use an open or closed circle? _________________
Would your arrow on the number line point to the left or to the right?_____
Answered by
GPT-4o mini
To solve the inequality and the equation:
1. **Solve the inequality**:
\( ax - 44 < 15 \)
Add 44 to both sides:
\( ax < 15 + 44 \)
\( ax < 59 \)
Divide both sides by \( a \) (assuming \( a > 0 \)):
\( x < \frac{59}{a} \)
If \( a < 0 \), the direction of the inequality would flip:
\( x > \frac{59}{a} \)
2. **Solve the equation**:
\( ax + 15 = 44 \)
Subtract 15 from both sides:
\( ax = 44 - 15 \)
\( ax = 29 \)
Divide both sides by \( a \) (assuming \( a > 0 \)):
\( x = \frac{29}{a} \)
If \( a < 0 \), it would be \( x = \frac{29}{a} \) as well, but \( a \) and \( x \) must be dealt with considering the sign.
### Summary of results:
- From the inequality, if \( a > 0 \), we find \( x < \frac{59}{a} \).
- From the equation, we find \( x = \frac{29}{a} \).
Considering the options:
1. \( x = 29a \)
2. \( x = \frac{29}{a} \)
3. \( x = \frac{59}{a} \)
4. \( x < \frac{59}{a} \)
5. \( x < 29a \)
6. \( x < 59a \)
7. \( x = 59a \)
8. \( x < \frac{29}{a} \)
The correct answers to the given inequality and equation would be:
- From the equation: \( x = \frac{29}{a} \) (option 2)
- From the inequality: \( x < \frac{59}{a} \) (option 4)
Thus, the answers are **2** and **4**.
1. **Solve the inequality**:
\( ax - 44 < 15 \)
Add 44 to both sides:
\( ax < 15 + 44 \)
\( ax < 59 \)
Divide both sides by \( a \) (assuming \( a > 0 \)):
\( x < \frac{59}{a} \)
If \( a < 0 \), the direction of the inequality would flip:
\( x > \frac{59}{a} \)
2. **Solve the equation**:
\( ax + 15 = 44 \)
Subtract 15 from both sides:
\( ax = 44 - 15 \)
\( ax = 29 \)
Divide both sides by \( a \) (assuming \( a > 0 \)):
\( x = \frac{29}{a} \)
If \( a < 0 \), it would be \( x = \frac{29}{a} \) as well, but \( a \) and \( x \) must be dealt with considering the sign.
### Summary of results:
- From the inequality, if \( a > 0 \), we find \( x < \frac{59}{a} \).
- From the equation, we find \( x = \frac{29}{a} \).
Considering the options:
1. \( x = 29a \)
2. \( x = \frac{29}{a} \)
3. \( x = \frac{59}{a} \)
4. \( x < \frac{59}{a} \)
5. \( x < 29a \)
6. \( x < 59a \)
7. \( x = 59a \)
8. \( x < \frac{29}{a} \)
The correct answers to the given inequality and equation would be:
- From the equation: \( x = \frac{29}{a} \) (option 2)
- From the inequality: \( x < \frac{59}{a} \) (option 4)
Thus, the answers are **2** and **4**.
Answered by
GPT-4o mini
Let's solve the equation step by step as you've outlined:
Given equation:
\[ 2(x - 3) + 21 = -3 \]
### Step 1: Distribute
**Distribute to get:**
\[ 2(x - 3) = 2x - 6 \]
Thus, the equation becomes:
\[ 2x - 6 + 21 = -3 \]
### Step 2: Combine like terms
**Combine like terms to get:**
\[ -6 + 21 = 15 \]
So the equation simplifies to:
\[ 2x + 15 = -3 \]
### Step 3: Use the (choose a property)
**Use the Addition Property of Equality** to isolate \( 2x \):
Subtract 15 from both sides:
\[ 2x + 15 - 15 = -3 - 15 \]
This gives us:
\[ 2x = -18 \]
### Step 4: Use the (choose a property)
**Use the Division Property of Equality** to solve for \( x \):
Divide both sides by 2:
\[ x = \frac{-18}{2} \]
So,
**Resulting answer:**
\[ x = -9 \]
### Final summary:
- Distribute to get: \( 2x - 6 + 21 = -3 \)
- Combine like terms to get: \( 2x + 15 = -3 \)
- Use the Addition Property of Equality to get: \( 2x = -18 \)
- Use the Division Property of Equality to get: \( x = -9 \)
Given equation:
\[ 2(x - 3) + 21 = -3 \]
### Step 1: Distribute
**Distribute to get:**
\[ 2(x - 3) = 2x - 6 \]
Thus, the equation becomes:
\[ 2x - 6 + 21 = -3 \]
### Step 2: Combine like terms
**Combine like terms to get:**
\[ -6 + 21 = 15 \]
So the equation simplifies to:
\[ 2x + 15 = -3 \]
### Step 3: Use the (choose a property)
**Use the Addition Property of Equality** to isolate \( 2x \):
Subtract 15 from both sides:
\[ 2x + 15 - 15 = -3 - 15 \]
This gives us:
\[ 2x = -18 \]
### Step 4: Use the (choose a property)
**Use the Division Property of Equality** to solve for \( x \):
Divide both sides by 2:
\[ x = \frac{-18}{2} \]
So,
**Resulting answer:**
\[ x = -9 \]
### Final summary:
- Distribute to get: \( 2x - 6 + 21 = -3 \)
- Combine like terms to get: \( 2x + 15 = -3 \)
- Use the Addition Property of Equality to get: \( 2x = -18 \)
- Use the Division Property of Equality to get: \( x = -9 \)
Answered by
GPT-4o mini
To solve the equation \( 3n + 9 + 4n = 2 \), we can break it down into the following steps:
### Step 1: Combine like terms
**Step 1: Combine like terms to get** \( 7n + 9 = 2 \)
(Here, \( 3n + 4n = 7n \)).
### Step 2: Use the Addition Property of Equality
**Step 2: Subtract 9 from both sides to get** \( 7n = 2 - 9 \)
(Here, we are removing the constant term from the left side).
### Step 3: Simplify and then use the Division Property of Equality
**Step 3: Simplify to get** \( 7n = -7 \)
**Step 3 continued: Divide both sides by 7 to get** \( n = -1 \).
### Final summary:
1. Combine like terms to get: \( 7n + 9 = 2 \)
2. Subtract 9 from both sides to get: \( 7n = -7 \)
3. Divide both sides by 7 to get: \( n = -1 \)
### Step 1: Combine like terms
**Step 1: Combine like terms to get** \( 7n + 9 = 2 \)
(Here, \( 3n + 4n = 7n \)).
### Step 2: Use the Addition Property of Equality
**Step 2: Subtract 9 from both sides to get** \( 7n = 2 - 9 \)
(Here, we are removing the constant term from the left side).
### Step 3: Simplify and then use the Division Property of Equality
**Step 3: Simplify to get** \( 7n = -7 \)
**Step 3 continued: Divide both sides by 7 to get** \( n = -1 \).
### Final summary:
1. Combine like terms to get: \( 7n + 9 = 2 \)
2. Subtract 9 from both sides to get: \( 7n = -7 \)
3. Divide both sides by 7 to get: \( n = -1 \)
Answered by
GPT-4o mini
To find which equation has the same first step as the equation \( 2x + 12 - 3x = 27 \), we first simplify \( 2x + 12 - 3x = 27 \) by combining like terms.
### The first step for our given equation:
1. Combine the \( 2x \) and \( -3x \):
\[
2x - 3x + 12 = 27 \implies -x + 12 = 27
\]
So, we are looking for an equation that has the same requirement to combine like terms.
### Now, let's analyze the options:
**A)** \( 4x - 17 + 2x = 9 \)
Combine \( 4x \) and \( 2x \):
\[
(4x + 2x) - 17 = 9 \implies 6x - 17 = 9
\]
(Yes, involves combining like terms)
**B)** \( 2x + 12 = -3x - 27 \)
This does not need combining like terms on the left side.
**C)** \( 2(x - 5) + 2x = 10 \)
First, distribute \( 2 \) in \( 2(x - 5) \):
\[
2x - 10 + 2x = 10 \implies 4x - 10 = 10
\]
(No direct combination of like terms)
**D)** \( 2x + 12 = 27 \)
This just requires moving \( +12 \) to the other side. No combining.
### Conclusion:
The only option that requires combining like terms in the same way (terms involving \( x \)) to solve is:
**A) \( 4x - 17 + 2x = 9 \)**.
So, **A** is the correct answer.
### The first step for our given equation:
1. Combine the \( 2x \) and \( -3x \):
\[
2x - 3x + 12 = 27 \implies -x + 12 = 27
\]
So, we are looking for an equation that has the same requirement to combine like terms.
### Now, let's analyze the options:
**A)** \( 4x - 17 + 2x = 9 \)
Combine \( 4x \) and \( 2x \):
\[
(4x + 2x) - 17 = 9 \implies 6x - 17 = 9
\]
(Yes, involves combining like terms)
**B)** \( 2x + 12 = -3x - 27 \)
This does not need combining like terms on the left side.
**C)** \( 2(x - 5) + 2x = 10 \)
First, distribute \( 2 \) in \( 2(x - 5) \):
\[
2x - 10 + 2x = 10 \implies 4x - 10 = 10
\]
(No direct combination of like terms)
**D)** \( 2x + 12 = 27 \)
This just requires moving \( +12 \) to the other side. No combining.
### Conclusion:
The only option that requires combining like terms in the same way (terms involving \( x \)) to solve is:
**A) \( 4x - 17 + 2x = 9 \)**.
So, **A** is the correct answer.
Answered by
GPT-4o mini
To solve the equation \( 7(5x + 2) = -4(6 - 5x) \) using the Distributive Property, we need to distribute both sides.
**Step 1: Distribute on the left side**
\[
7(5x) + 7(2) = 35x + 14
\]
**Step 2: Distribute on the right side**
\[
-4(6) + (-4)(-5x) = -24 + 20x
\]
Now, putting both sides together after distributing gives us:
\[
35x + 14 = -24 + 20x
\]
### Now let's look at the options you provided:
A) \( 35x + 14 = -24 + 20x \)
(This matches our derived equation.)
B) \( 35x + 2 = -24 - 5x \)
(This does not match; incorrect constants and coefficients.)
C) \( 5x + 14 = 6 + 20x \)
(This does not match; incorrect coefficients and constants.)
D) \( 35x + 2 = 24 - 20x \)
(This does not match; incorrect constants and signs.)
E) \( 35x + 14 = -24 - 20x \)
(This does not match; while it has the correct left side, the right side is incorrect because of the sign in front of \( 20x \).)
### Conclusion:
The correct choice that results from using the Distributive Property on the given equation is:
**A)** \( 35x + 14 = -24 + 20x \)
**Step 1: Distribute on the left side**
\[
7(5x) + 7(2) = 35x + 14
\]
**Step 2: Distribute on the right side**
\[
-4(6) + (-4)(-5x) = -24 + 20x
\]
Now, putting both sides together after distributing gives us:
\[
35x + 14 = -24 + 20x
\]
### Now let's look at the options you provided:
A) \( 35x + 14 = -24 + 20x \)
(This matches our derived equation.)
B) \( 35x + 2 = -24 - 5x \)
(This does not match; incorrect constants and coefficients.)
C) \( 5x + 14 = 6 + 20x \)
(This does not match; incorrect coefficients and constants.)
D) \( 35x + 2 = 24 - 20x \)
(This does not match; incorrect constants and signs.)
E) \( 35x + 14 = -24 - 20x \)
(This does not match; while it has the correct left side, the right side is incorrect because of the sign in front of \( 20x \).)
### Conclusion:
The correct choice that results from using the Distributive Property on the given equation is:
**A)** \( 35x + 14 = -24 + 20x \)
Answered by
GPT-4o mini
Let's simplify the given equation step by step.
### Original Equation
\[ 2(x - 5) = 9 - 3x + 6 + 8 + 3x + 7 \]
### Simplify the Left Side
**Left Side: Using the Distributive Property**
\[ 2(x - 5) = 2x - 10 \]
So, the left side simplifies to:
\[ 2x - 10 \]
### Simplify the Right Side
**Right Side: Combining Like Terms**
First, let's combine the constant terms and simplify:
1. Identify the constant terms: \( 9 + 6 + 8 + 7 \)
2. Combine them:
\[ 9 + 6 = 15 \]
\[ 15 + 8 = 23 \]
\[ 23 + 7 = 30 \]
Now, let's see how the terms combine:
- The \( -3x \) and \( +3x \) cancel each other out (i.e., \( -3x + 3x = 0 \))
So, the right side simplifies to:
\[ 30 \]
### Final Summary
- **Simplify the left side of the equation:** \( 2x - 10 \)
- **Simplify the right side of the equation:** \( 30 \)
Putting it all together, we have:
- **Left Side Simplified:** \( 2x - 10 \)
- **Right Side Simplified:** \( 30 \)
### Original Equation
\[ 2(x - 5) = 9 - 3x + 6 + 8 + 3x + 7 \]
### Simplify the Left Side
**Left Side: Using the Distributive Property**
\[ 2(x - 5) = 2x - 10 \]
So, the left side simplifies to:
\[ 2x - 10 \]
### Simplify the Right Side
**Right Side: Combining Like Terms**
First, let's combine the constant terms and simplify:
1. Identify the constant terms: \( 9 + 6 + 8 + 7 \)
2. Combine them:
\[ 9 + 6 = 15 \]
\[ 15 + 8 = 23 \]
\[ 23 + 7 = 30 \]
Now, let's see how the terms combine:
- The \( -3x \) and \( +3x \) cancel each other out (i.e., \( -3x + 3x = 0 \))
So, the right side simplifies to:
\[ 30 \]
### Final Summary
- **Simplify the left side of the equation:** \( 2x - 10 \)
- **Simplify the right side of the equation:** \( 30 \)
Putting it all together, we have:
- **Left Side Simplified:** \( 2x - 10 \)
- **Right Side Simplified:** \( 30 \)
Answered by
GPT-4o mini
Let's solve the equation step by step and fill in the blanks:
Given equation:
\[ 3(x - 5) + 2x = 0 \]
### Step 1: Distribute
Distribute \( 3 \) to \( (x - 5) \):
\[ 3x - 15 + 2x = 0 \]
### Filling in the blanks
So the equation now looks like:
**3x +** **(-15)** **+ 2x = 0**
### Step 2: Combine like terms
Combine \( 3x \) and \( 2x \):
\[ 5x - 15 = 0 \]
Now, we can fill in the second blank:
**5x =** **15**
(We moved \( -15 \) to the other side of the equation.)
### Step 3: Solve for \( x \)
Now, divide both sides by \( 5 \):
\[ x = \frac{15}{5} = 3 \]
Finally, we fill in the last blank:
**x =** **3**
### Summary of the filled steps:
- 3x + **(-15)** + 2x = 0
- 5x = **15**
- x = **3**
Given equation:
\[ 3(x - 5) + 2x = 0 \]
### Step 1: Distribute
Distribute \( 3 \) to \( (x - 5) \):
\[ 3x - 15 + 2x = 0 \]
### Filling in the blanks
So the equation now looks like:
**3x +** **(-15)** **+ 2x = 0**
### Step 2: Combine like terms
Combine \( 3x \) and \( 2x \):
\[ 5x - 15 = 0 \]
Now, we can fill in the second blank:
**5x =** **15**
(We moved \( -15 \) to the other side of the equation.)
### Step 3: Solve for \( x \)
Now, divide both sides by \( 5 \):
\[ x = \frac{15}{5} = 3 \]
Finally, we fill in the last blank:
**x =** **3**
### Summary of the filled steps:
- 3x + **(-15)** + 2x = 0
- 5x = **15**
- x = **3**
Answered by
GPT-4o mini
Let's start by simplifying the equation \( 3 - 15x + 24 + 16x = 4x - 24 - 4x \).
### Step 1: Combine Like Terms
For the left side:
- Combine constant terms: \( 3 + 24 = 27 \)
- Combine the \( x \) terms: \( -15x + 16x = x \)
So, the left side becomes:
\[ 27 + x \]
For the right side:
- Combine constant terms: \( -24 - 4x \) simplifies to \( -24 - 4x + 4x = -24 \).
So, the right side becomes:
\[ 4x - 4x - 24 = -24 \]
### Resulting Simplified Equation:
The equation can now be rewritten as:
\[ 27 + x = -24 \]
### Step 2: Solving for \( x \)
To isolate \( x \), we need to perform operations. The best next step would be to move \( 27 \) to the right side.
- Subtract \( 27 \) from both sides of the equation.
### Conclusion:
Thus, the correct answer is:
**A). Subtract 24 from both sides of the equation.**
### Given the choice of options, it seems like there may have been a small error in the available answers since the appropriate step we identified is to **subtract 27 (not 24)** from both sides but **not among the choices**.
If the initial equation is kept as is and focusing on what's provided, I recommend using **C). Subtract 27 from both sides of the equation** as you would typically be isolating a variable from a sum.
### Step 1: Combine Like Terms
For the left side:
- Combine constant terms: \( 3 + 24 = 27 \)
- Combine the \( x \) terms: \( -15x + 16x = x \)
So, the left side becomes:
\[ 27 + x \]
For the right side:
- Combine constant terms: \( -24 - 4x \) simplifies to \( -24 - 4x + 4x = -24 \).
So, the right side becomes:
\[ 4x - 4x - 24 = -24 \]
### Resulting Simplified Equation:
The equation can now be rewritten as:
\[ 27 + x = -24 \]
### Step 2: Solving for \( x \)
To isolate \( x \), we need to perform operations. The best next step would be to move \( 27 \) to the right side.
- Subtract \( 27 \) from both sides of the equation.
### Conclusion:
Thus, the correct answer is:
**A). Subtract 24 from both sides of the equation.**
### Given the choice of options, it seems like there may have been a small error in the available answers since the appropriate step we identified is to **subtract 27 (not 24)** from both sides but **not among the choices**.
If the initial equation is kept as is and focusing on what's provided, I recommend using **C). Subtract 27 from both sides of the equation** as you would typically be isolating a variable from a sum.
Answered by
GPT-4o mini
The speed limit sign states that the speed limit is 55 mph. This means that you can drive at speeds that are **equal to or less than** 55 mph.
To represent this in an inequality:
- You can drive at speeds that are **less than or equal to** 55 mph, which can be expressed as:
\[
x \leq 55
\]
Therefore, the correct inequality that represents all speeds you can legally drive on the expressway is:
**x ≤ 55**
To represent this in an inequality:
- You can drive at speeds that are **less than or equal to** 55 mph, which can be expressed as:
\[
x \leq 55
\]
Therefore, the correct inequality that represents all speeds you can legally drive on the expressway is:
**x ≤ 55**
Answered by
GPT-4o mini
To graph the inequality \( 35 < 7x \), we can start by rewriting it in a more standard form for better understanding:
1. Rearranging gives us:
\[ 7x > 35 \]
2. Dividing both sides by 7:
\[ x > 5 \]
### Graphing the Inequality:
1. **Open or Closed Circle?**
- Since the inequality is **strictly greater than** (not inclusive), we use an **open circle** to indicate that 5 is not included in the solution.
2. **Direction of the Arrow?**
- Because \( x \) is greater than 5, the arrow on the number line will point to the **right**, indicating that all values greater than 5 are included in the solution.
### Summary:
- **Open or closed circle?**: **Open circle**
- **Arrow direction?**: **Right**
1. Rearranging gives us:
\[ 7x > 35 \]
2. Dividing both sides by 7:
\[ x > 5 \]
### Graphing the Inequality:
1. **Open or Closed Circle?**
- Since the inequality is **strictly greater than** (not inclusive), we use an **open circle** to indicate that 5 is not included in the solution.
2. **Direction of the Arrow?**
- Because \( x \) is greater than 5, the arrow on the number line will point to the **right**, indicating that all values greater than 5 are included in the solution.
### Summary:
- **Open or closed circle?**: **Open circle**
- **Arrow direction?**: **Right**
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