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Original Question
Determine which of the following statements is/are true: #1: 3⋅20=4(6+9) #2: 4×12=6×(8×2)12+6⋅64−−√=2⋅4⋅8 #3: 4⋅18−4⋅6=42+4⋅8 R...Asked by Yes.
Determine which of the following statements is/are true:
#1: 3⋅20=4(6+9)
#2: 4×12=6×(8×2)12+6⋅64−−√=2⋅4⋅8
#3: 4⋅18−4⋅6=4^2+4⋅8
Responses:
1 and 2
2 and 3
1
1 and 3
#1: 3⋅20=4(6+9)
#2: 4×12=6×(8×2)12+6⋅64−−√=2⋅4⋅8
#3: 4⋅18−4⋅6=4^2+4⋅8
Responses:
1 and 2
2 and 3
1
1 and 3
Answers
Answered by
Yes.
Which expression is A, in order to have a true equation A=5−113?(1 point)
Responses:
A= 1 2/3+13
A= 1 5/3+13
A= −1 2/3+13
A= 1 2/3−13
Responses:
A= 1 2/3+13
A= 1 5/3+13
A= −1 2/3+13
A= 1 2/3−13
Answered by
Yes.
Consider the following expressions:
−6+3⋅5 ______ 16−−√+32÷8 .
Which operator can be inserted to make the statement true?
Responses:
≤
=
<
>
−6+3⋅5 ______ 16−−√+32÷8 .
Which operator can be inserted to make the statement true?
Responses:
≤
=
<
>
Answered by
Yes.
When the value of r is 23, the equation 3r−47.5=21.5 is a true statement. Which of the following values of r makes the inequality 3r−47.5>21.5 a true statement?
Responses:
23.5
23
22.5
22
Responses:
23.5
23
22.5
22
Answered by
Yes.
Which of the following values for x makes the inequality 4x+1−−−−−√ ≤5.83 true?(1 point)
Responses:
9
8
9.5
10
Responses:
9
8
9.5
10
Answered by
GPT-4o mini
Let's analyze each statement one by one to determine which are true.
**Statement #1:**
\[ 3 \cdot 20 = 4(6 + 9) \]
Calculating the left side:
\[ 3 \cdot 20 = 60 \]
Calculating the right side:
\[ 6 + 9 = 15 \]
\[ 4 \cdot 15 = 60 \]
Both sides are equal, so this statement is **true**.
---
**Statement #2:**
\[ 4 \times 12 = 6 \times (8 \times 2)12 + 6 \cdot \sqrt{64} = 2 \cdot 4 \cdot 8 \]
Calculating the left side:
\[ 4 \cdot 12 = 48 \]
Now for the right side:
Start with \( 6 \times (8 \times 2) = 6 \times 16 = 96 \). We will just write \( 12 \) at the end for simplicity in verifying.
Next, \( 6 \cdot \sqrt{64} = 6 \cdot 8 = 48 \).
So we have:
\[ 96 + 48 = 144 \]
Also, the expression \( 2 \cdot 4 \cdot 8 = 64 \).
So the total is not equal:
Left side is 48, right side simplifies to 144 or 64 depending on interpretation, hence this entire statement is **false**.
---
**Statement #3:**
\[ 4 \cdot 18 - 4 \cdot 6 = 4^2 + 4 \cdot 8 \]
Calculating the left side:
\[ 4 \cdot 18 = 72 \]
\[ 4 \cdot 6 = 24 \]
So:
\[ 72 - 24 = 48 \]
Calculating the right side:
\[ 4^2 = 16 \]
\[ 4 \cdot 8 = 32 \]
So:
\[ 16 + 32 = 48 \]
Both sides are equal, so this statement is **true**.
---
**Conclusion:**
Statements #1 and #3 are true, while Statement #2 is false.
Thus, the correct response is: **1 and 3**.
**Statement #1:**
\[ 3 \cdot 20 = 4(6 + 9) \]
Calculating the left side:
\[ 3 \cdot 20 = 60 \]
Calculating the right side:
\[ 6 + 9 = 15 \]
\[ 4 \cdot 15 = 60 \]
Both sides are equal, so this statement is **true**.
---
**Statement #2:**
\[ 4 \times 12 = 6 \times (8 \times 2)12 + 6 \cdot \sqrt{64} = 2 \cdot 4 \cdot 8 \]
Calculating the left side:
\[ 4 \cdot 12 = 48 \]
Now for the right side:
Start with \( 6 \times (8 \times 2) = 6 \times 16 = 96 \). We will just write \( 12 \) at the end for simplicity in verifying.
Next, \( 6 \cdot \sqrt{64} = 6 \cdot 8 = 48 \).
So we have:
\[ 96 + 48 = 144 \]
Also, the expression \( 2 \cdot 4 \cdot 8 = 64 \).
So the total is not equal:
Left side is 48, right side simplifies to 144 or 64 depending on interpretation, hence this entire statement is **false**.
---
**Statement #3:**
\[ 4 \cdot 18 - 4 \cdot 6 = 4^2 + 4 \cdot 8 \]
Calculating the left side:
\[ 4 \cdot 18 = 72 \]
\[ 4 \cdot 6 = 24 \]
So:
\[ 72 - 24 = 48 \]
Calculating the right side:
\[ 4^2 = 16 \]
\[ 4 \cdot 8 = 32 \]
So:
\[ 16 + 32 = 48 \]
Both sides are equal, so this statement is **true**.
---
**Conclusion:**
Statements #1 and #3 are true, while Statement #2 is false.
Thus, the correct response is: **1 and 3**.
Answered by
GPT-4o mini
To find the expression \( A \) that makes the equation \( A = 5 - \frac{1}{13} \) true, we can first calculate the value of \( 5 - \frac{1}{13} \).
1. Convert \( 5 \) to a fraction with a denominator of \( 13 \):
\[
5 = \frac{65}{13}
\]
2. Now, subtract \( \frac{1}{13} \):
\[
5 - \frac{1}{13} = \frac{65}{13} - \frac{1}{13} = \frac{64}{13}
\]
Now, we need to simplify \( \frac{64}{13} \):
\[
\frac{64}{13} = 4 \frac{12}{13}
\]
Next, we can analyze the responses given to check for equivalence:
1. **A = \( 1 \frac{2}{3} + 13 \)**
- \( 1 \frac{2}{3} = \frac{5}{3} \). Thus, \( 1 \frac{2}{3} + 13 = \frac{5}{3} + 13 \) which does not equal \( \frac{64}{13} \).
2. **A = \( 1 \frac{5}{3} + 13 \)**
- \( 1 \frac{5}{3} = \frac{8}{3} \). Thus, \( \frac{8}{3} + 13 = \frac{8}{3} + \frac{39}{3} = \frac{47}{3} \), which does not equal \( \frac{64}{13} \).
3. **A = \( -1 \frac{2}{3} + 13 \)**
- \( -1 \frac{2}{3} = -\frac{5}{3} \). Thus, \( -1 \frac{2}{3} + 13 = -\frac{5}{3} + 13 = -\frac{5}{3} + \frac{39}{3} = \frac{34}{3} \), which does not equal \( \frac{64}{13} \).
4. **A = \( 1 \frac{2}{3} - 13 \)**
- \( 1 \frac{2}{3} = \frac{5}{3} \). Thus, \( 1 \frac{2}{3} - 13 = \frac{5}{3} - 13 = \frac{5}{3} - \frac{39}{3} = -\frac{34}{3} \), which does not equal \( \frac{64}{13} \).
None of the provided expressions yield a value equal to \( 5 - \frac{1}{13} \) which we found to be \( \frac{64}{13} \).
I apologize for the oversight. I realize that I have not suspected the mistake lies in the expressions you provided. These expressions seem not equivalent to \( 5 - \frac{1}{13}\) i.e. \( \frac{64}{13}\).
Therefore, none of the responses correctly satisfy \( A = 5 - \frac{1}{13} \). Please check the answers or consider that there might be an error in the problem statement.
1. Convert \( 5 \) to a fraction with a denominator of \( 13 \):
\[
5 = \frac{65}{13}
\]
2. Now, subtract \( \frac{1}{13} \):
\[
5 - \frac{1}{13} = \frac{65}{13} - \frac{1}{13} = \frac{64}{13}
\]
Now, we need to simplify \( \frac{64}{13} \):
\[
\frac{64}{13} = 4 \frac{12}{13}
\]
Next, we can analyze the responses given to check for equivalence:
1. **A = \( 1 \frac{2}{3} + 13 \)**
- \( 1 \frac{2}{3} = \frac{5}{3} \). Thus, \( 1 \frac{2}{3} + 13 = \frac{5}{3} + 13 \) which does not equal \( \frac{64}{13} \).
2. **A = \( 1 \frac{5}{3} + 13 \)**
- \( 1 \frac{5}{3} = \frac{8}{3} \). Thus, \( \frac{8}{3} + 13 = \frac{8}{3} + \frac{39}{3} = \frac{47}{3} \), which does not equal \( \frac{64}{13} \).
3. **A = \( -1 \frac{2}{3} + 13 \)**
- \( -1 \frac{2}{3} = -\frac{5}{3} \). Thus, \( -1 \frac{2}{3} + 13 = -\frac{5}{3} + 13 = -\frac{5}{3} + \frac{39}{3} = \frac{34}{3} \), which does not equal \( \frac{64}{13} \).
4. **A = \( 1 \frac{2}{3} - 13 \)**
- \( 1 \frac{2}{3} = \frac{5}{3} \). Thus, \( 1 \frac{2}{3} - 13 = \frac{5}{3} - 13 = \frac{5}{3} - \frac{39}{3} = -\frac{34}{3} \), which does not equal \( \frac{64}{13} \).
None of the provided expressions yield a value equal to \( 5 - \frac{1}{13} \) which we found to be \( \frac{64}{13} \).
I apologize for the oversight. I realize that I have not suspected the mistake lies in the expressions you provided. These expressions seem not equivalent to \( 5 - \frac{1}{13}\) i.e. \( \frac{64}{13}\).
Therefore, none of the responses correctly satisfy \( A = 5 - \frac{1}{13} \). Please check the answers or consider that there might be an error in the problem statement.
Answered by
GPT-4o mini
Let's simplify both sides of the expression to determine which operator can be inserted to make the statement true:
### Left Side:
\[
-6 + 3 \cdot 5
\]
Calculating:
\[
3 \cdot 5 = 15
\]
So,
\[
-6 + 15 = 9
\]
### Right Side:
\[
16 - \sqrt{16} + 32 \div 8
\]
Calculating:
1. Find the square root:
\[
\sqrt{16} = 4
\]
2. Perform the division:
\[
32 \div 8 = 4
\]
3. Now substitute back into the expression:
\[
16 - 4 + 4
\]
Calculating step-by-step:
\[
16 - 4 = 12
\]
and then,
\[
12 + 4 = 16
\]
### Summary of Results:
- Left side evaluates to \( 9 \).
- Right side evaluates to \( 16 \).
### Compare Values:
Now we can compare \( 9 \) and \( 16 \) using the provided operators:
- \( 9 \leq 16 \) is **true**.
- \( 9 = 16 \) is **false**.
- \( 9 < 16 \) is **true**.
- \( 9 > 16 \) is **false**.
### Possible Operators:
The valid operators that can be inserted to make the statement true are:
- \( \leq \)
- \( < \)
Thus, both \( \leq \) and \( < \) can be inserted. However, if you are asked to choose the most common one for the equation, it would be \( < \).
Final answer: **<** (the operator that can be inserted to make the statement true).
### Left Side:
\[
-6 + 3 \cdot 5
\]
Calculating:
\[
3 \cdot 5 = 15
\]
So,
\[
-6 + 15 = 9
\]
### Right Side:
\[
16 - \sqrt{16} + 32 \div 8
\]
Calculating:
1. Find the square root:
\[
\sqrt{16} = 4
\]
2. Perform the division:
\[
32 \div 8 = 4
\]
3. Now substitute back into the expression:
\[
16 - 4 + 4
\]
Calculating step-by-step:
\[
16 - 4 = 12
\]
and then,
\[
12 + 4 = 16
\]
### Summary of Results:
- Left side evaluates to \( 9 \).
- Right side evaluates to \( 16 \).
### Compare Values:
Now we can compare \( 9 \) and \( 16 \) using the provided operators:
- \( 9 \leq 16 \) is **true**.
- \( 9 = 16 \) is **false**.
- \( 9 < 16 \) is **true**.
- \( 9 > 16 \) is **false**.
### Possible Operators:
The valid operators that can be inserted to make the statement true are:
- \( \leq \)
- \( < \)
Thus, both \( \leq \) and \( < \) can be inserted. However, if you are asked to choose the most common one for the equation, it would be \( < \).
Final answer: **<** (the operator that can be inserted to make the statement true).
Answered by
GPT-4o mini
Let's first rewrite the inequality \(3r - 47.5 > 21.5\) and solve for \(r\).
### Step 1: Solve the Inequality
1. Start with the inequality:
\[
3r - 47.5 > 21.5
\]
2. Add \(47.5\) to both sides:
\[
3r > 21.5 + 47.5
\]
\[
3r > 69
\]
3. Now, divide both sides by \(3\):
\[
r > \frac{69}{3}
\]
\[
r > 23
\]
### Step 2: Determine Which Values Are Greater Than 23
Now that we know \(r\) must be greater than \(23\), let's check the provided options:
1. **23.5** (is greater than \(23\))
2. **23** (is NOT greater than \(23\))
3. **22.5** (is NOT greater than \(23\))
4. **22** (is NOT greater than \(23\))
### Conclusion
The only value among the options that satisfies \(r > 23\) is:
- **23.5**
Thus, the value of \(r\) that makes the inequality \(3r - 47.5 > 21.5\) a true statement is **23.5**.
### Step 1: Solve the Inequality
1. Start with the inequality:
\[
3r - 47.5 > 21.5
\]
2. Add \(47.5\) to both sides:
\[
3r > 21.5 + 47.5
\]
\[
3r > 69
\]
3. Now, divide both sides by \(3\):
\[
r > \frac{69}{3}
\]
\[
r > 23
\]
### Step 2: Determine Which Values Are Greater Than 23
Now that we know \(r\) must be greater than \(23\), let's check the provided options:
1. **23.5** (is greater than \(23\))
2. **23** (is NOT greater than \(23\))
3. **22.5** (is NOT greater than \(23\))
4. **22** (is NOT greater than \(23\))
### Conclusion
The only value among the options that satisfies \(r > 23\) is:
- **23.5**
Thus, the value of \(r\) that makes the inequality \(3r - 47.5 > 21.5\) a true statement is **23.5**.
Answered by
GPT-4o mini
To determine which value for \( x \) makes the inequality \( \sqrt{4x + 1} \leq 5.83 \) true, we can start by squaring both sides of the inequality to eliminate the square root.
### Step 1: Set Up the Inequality
\[
\sqrt{4x + 1} \leq 5.83
\]
### Step 2: Square Both Sides
\[
4x + 1 \leq (5.83)^2
\]
Calculating \( (5.83)^2 \):
\[
(5.83)^2 = 33.9889
\]
### Step 3: Solve the Inequality
Now we have:
\[
4x + 1 \leq 33.9889
\]
Subtract \( 1 \) from both sides:
\[
4x \leq 32.9889
\]
Divide by \( 4 \):
\[
x \leq \frac{32.9889}{4}
\]
\[
x \leq 8.247225
\]
### Step 4: List the Responses
Now let's check which of the given responses is less than or equal to \( 8.247225 \):
1. \( 9 \) (not valid, since \( 9 > 8.247225 \))
2. \( 8 \) (valid, since \( 8 \leq 8.247225 \))
3. \( 9.5 \) (not valid, since \( 9.5 > 8.247225 \))
4. \( 10 \) (not valid, since \( 10 > 8.247225 \))
### Conclusion
The value for \( x \) that makes the inequality \( \sqrt{4x + 1} \leq 5.83 \) true is **8**.
### Step 1: Set Up the Inequality
\[
\sqrt{4x + 1} \leq 5.83
\]
### Step 2: Square Both Sides
\[
4x + 1 \leq (5.83)^2
\]
Calculating \( (5.83)^2 \):
\[
(5.83)^2 = 33.9889
\]
### Step 3: Solve the Inequality
Now we have:
\[
4x + 1 \leq 33.9889
\]
Subtract \( 1 \) from both sides:
\[
4x \leq 32.9889
\]
Divide by \( 4 \):
\[
x \leq \frac{32.9889}{4}
\]
\[
x \leq 8.247225
\]
### Step 4: List the Responses
Now let's check which of the given responses is less than or equal to \( 8.247225 \):
1. \( 9 \) (not valid, since \( 9 > 8.247225 \))
2. \( 8 \) (valid, since \( 8 \leq 8.247225 \))
3. \( 9.5 \) (not valid, since \( 9.5 > 8.247225 \))
4. \( 10 \) (not valid, since \( 10 > 8.247225 \))
### Conclusion
The value for \( x \) that makes the inequality \( \sqrt{4x + 1} \leq 5.83 \) true is **8**.