Asked by jimmy.
Consider the following statement:
8⋅3+2⋅6
8
⋅
3
+
2
⋅
6
______60÷5⋅3
60
÷
5
⋅
3
.
Which operator can be inserted to make the statement true?
(1 point)
Responses
=
=
equals
≠
≠
does not equal
>
>
greater than
<
8⋅3+2⋅6
8
⋅
3
+
2
⋅
6
______60÷5⋅3
60
÷
5
⋅
3
.
Which operator can be inserted to make the statement true?
(1 point)
Responses
=
=
equals
≠
≠
does not equal
>
>
greater than
<
Answers
Answered by
jimmy.
Which of the following values of x makes the equation 2x−−√+13=27
2
𝑥
+
13
=
27
true?(1 point)
Responses
16
16
49
49
25
25
9
2
𝑥
+
13
=
27
true?(1 point)
Responses
16
16
49
49
25
25
9
Answered by
jimmy.
Which option contains the solution set for the value of k in the equation 7+2k2=25
7
+
2
𝑘
2
=
25
?(1 point)
Responses
{−3,3}
{
−
3
,
3
}
left brace negative 3 comma 3 right brace
∅
∅
empty set
{3}
{
3
}
left brace 3 right brace
{−9,9}
7
+
2
𝑘
2
=
25
?(1 point)
Responses
{−3,3}
{
−
3
,
3
}
left brace negative 3 comma 3 right brace
∅
∅
empty set
{3}
{
3
}
left brace 3 right brace
{−9,9}
Answered by
jimmy.
Which option is the set that expresses the solution to the inequality k+1<−3
𝑘
+
1
<
−
3
?(1 point)
Responses
{k:k>−4}
{
𝑘
:
𝑘
>
−
4
}
left brace k colon k greater than negative 4 right brace
{k:k<−4}
{
𝑘
:
𝑘
<
−
4
}
left brace k colon k less than negative 4 right brace
{k:k>4}
{
𝑘
:
𝑘
>
4
}
left brace k colon k greater than 4 right brace
{k:k<4}
𝑘
+
1
<
−
3
?(1 point)
Responses
{k:k>−4}
{
𝑘
:
𝑘
>
−
4
}
left brace k colon k greater than negative 4 right brace
{k:k<−4}
{
𝑘
:
𝑘
<
−
4
}
left brace k colon k less than negative 4 right brace
{k:k>4}
{
𝑘
:
𝑘
>
4
}
left brace k colon k greater than 4 right brace
{k:k<4}
Answered by
jimmy.
How many solutions will the equation 4h+6=14−12(16−8h)
4
ℎ
+
6
=
14
−
1
2
(
16
−
8
ℎ
)
have?(1 point)
Responses
no solution
no solution
infinite/many solutions
infinite/many solutions
two solutions
two solutions
one solution
4
ℎ
+
6
=
14
−
1
2
(
16
−
8
ℎ
)
have?(1 point)
Responses
no solution
no solution
infinite/many solutions
infinite/many solutions
two solutions
two solutions
one solution
Answered by
jimmy.
wrong
Answered by
GPT-4o mini
To determine which operator can be inserted to make the statement true, we first need to compute both sides of the equation.
**Left side:**
\[ 8 \cdot 3 + 2 \cdot 6 \]
- Calculating \( 8 \cdot 3 = 24 \)
- Calculating \( 2 \cdot 6 = 12 \)
- Adding those together: \( 24 + 12 = 36 \)
So, the left side evaluates to \( 36 \).
**Right side:**
\[ 60 \div 5 \cdot 3 \]
- First calculating \( 60 \div 5 = 12 \)
- Then calculating \( 12 \cdot 3 = 36 \)
So, the right side evaluates to \( 36 \).
Now we can see that both sides are equal:
\[ 36 = 36 \]
Therefore, the correct operator to insert is:
\[ \text{=} \]
The statement can be made true by inserting the equals sign.
**Left side:**
\[ 8 \cdot 3 + 2 \cdot 6 \]
- Calculating \( 8 \cdot 3 = 24 \)
- Calculating \( 2 \cdot 6 = 12 \)
- Adding those together: \( 24 + 12 = 36 \)
So, the left side evaluates to \( 36 \).
**Right side:**
\[ 60 \div 5 \cdot 3 \]
- First calculating \( 60 \div 5 = 12 \)
- Then calculating \( 12 \cdot 3 = 36 \)
So, the right side evaluates to \( 36 \).
Now we can see that both sides are equal:
\[ 36 = 36 \]
Therefore, the correct operator to insert is:
\[ \text{=} \]
The statement can be made true by inserting the equals sign.
Answered by
GPT-4o mini
To solve the equation \( \sqrt{2x} + 13 = 27 \), we will start by isolating the square root:
1. Subtract 13 from both sides:
\[
\sqrt{2x} = 27 - 13
\]
\[
\sqrt{2x} = 14
\]
2. Next, square both sides to eliminate the square root:
\[
2x = 14^2
\]
\[
2x = 196
\]
3. Divide both sides by 2 to solve for \( x \):
\[
x = \frac{196}{2}
\]
\[
x = 98
\]
Now, let's check which of the provided options gives a true statement when plugged back into the original equation:
- **For \( x = 16 \)**:
\[
\sqrt{2 \cdot 16} + 13 = \sqrt{32} + 13 \approx 5.66 + 13 = 18.66 \quad \text{(not equal to 27)}
\]
- **For \( x = 49 \)**:
\[
\sqrt{2 \cdot 49} + 13 = \sqrt{98} + 13 \approx 9.898 + 13 = 22.898 \quad \text{(not equal to 27)}
\]
- **For \( x = 25 \)**:
\[
\sqrt{2 \cdot 25} + 13 = \sqrt{50} + 13 \approx 7.071 + 13 = 20.071 \quad \text{(not equal to 27)}
\]
- **For \( x = 9 \)**:
\[
\sqrt{2 \cdot 9} + 13 = \sqrt{18} + 13 \approx 4.242 + 13 = 17.242 \quad \text{(not equal to 27)}
\]
None of the provided values satisfy the equation. Based on my calculations, the true value that satisfies the equation \( \sqrt{2x} + 13 = 27 \) is \( x = 98 \), which is not listed among the options.
1. Subtract 13 from both sides:
\[
\sqrt{2x} = 27 - 13
\]
\[
\sqrt{2x} = 14
\]
2. Next, square both sides to eliminate the square root:
\[
2x = 14^2
\]
\[
2x = 196
\]
3. Divide both sides by 2 to solve for \( x \):
\[
x = \frac{196}{2}
\]
\[
x = 98
\]
Now, let's check which of the provided options gives a true statement when plugged back into the original equation:
- **For \( x = 16 \)**:
\[
\sqrt{2 \cdot 16} + 13 = \sqrt{32} + 13 \approx 5.66 + 13 = 18.66 \quad \text{(not equal to 27)}
\]
- **For \( x = 49 \)**:
\[
\sqrt{2 \cdot 49} + 13 = \sqrt{98} + 13 \approx 9.898 + 13 = 22.898 \quad \text{(not equal to 27)}
\]
- **For \( x = 25 \)**:
\[
\sqrt{2 \cdot 25} + 13 = \sqrt{50} + 13 \approx 7.071 + 13 = 20.071 \quad \text{(not equal to 27)}
\]
- **For \( x = 9 \)**:
\[
\sqrt{2 \cdot 9} + 13 = \sqrt{18} + 13 \approx 4.242 + 13 = 17.242 \quad \text{(not equal to 27)}
\]
None of the provided values satisfy the equation. Based on my calculations, the true value that satisfies the equation \( \sqrt{2x} + 13 = 27 \) is \( x = 98 \), which is not listed among the options.
Answered by
GPT-4o mini
To solve the equation \( 7 + 2k^2 = 25 \), we can follow these steps:
1. Subtract 7 from both sides:
\[
2k^2 = 25 - 7
\]
\[
2k^2 = 18
\]
2. Divide both sides by 2:
\[
k^2 = \frac{18}{2}
\]
\[
k^2 = 9
\]
3. Take the square root of both sides:
\[
k = \pm \sqrt{9}
\]
\[
k = \pm 3
\]
So, the solutions for \( k \) are \( k = 3 \) and \( k = -3 \).
The solution set is:
\[
\{ -3, 3 \}
\]
Hence, the correct option is:
\[
\{ -3, 3 \}
\]
1. Subtract 7 from both sides:
\[
2k^2 = 25 - 7
\]
\[
2k^2 = 18
\]
2. Divide both sides by 2:
\[
k^2 = \frac{18}{2}
\]
\[
k^2 = 9
\]
3. Take the square root of both sides:
\[
k = \pm \sqrt{9}
\]
\[
k = \pm 3
\]
So, the solutions for \( k \) are \( k = 3 \) and \( k = -3 \).
The solution set is:
\[
\{ -3, 3 \}
\]
Hence, the correct option is:
\[
\{ -3, 3 \}
\]
Answered by
GPT-4o mini
To solve the inequality \( k + 1 < -3 \), we can follow these steps:
1. Subtract 1 from both sides:
\[
k < -3 - 1
\]
\[
k < -4
\]
The solution to the inequality is \( k < -4 \).
Thus, the correct option that expresses this solution set is:
\[
\{ k : k < -4 \}
\]
1. Subtract 1 from both sides:
\[
k < -3 - 1
\]
\[
k < -4
\]
The solution to the inequality is \( k < -4 \).
Thus, the correct option that expresses this solution set is:
\[
\{ k : k < -4 \}
\]
Answered by
GPT-4o mini
To determine how many solutions the equation \( 4h + 6 = 14 - 12(16 - 8h) \) has, we can start by simplifying both sides of the equation.
1. Begin with the original equation:
\[
4h + 6 = 14 - 12(16 - 8h)
\]
2. Distribute \(-12\) on the right side:
\[
4h + 6 = 14 - 192 + 96h
\]
\[
4h + 6 = -178 + 96h
\]
3. Rearrange the equation to collect all terms involving \(h\) on one side and constant terms on the other side:
\[
4h - 96h = -178 - 6
\]
\[
-92h = -184
\]
4. Divide both sides by \(-92\):
\[
h = \frac{-184}{-92} = 2
\]
Since we obtained a specific numeric value for \(h\), there is exactly **one solution** to the equation.
Thus, the correct response is:
\[
\text{one solution}
\]
1. Begin with the original equation:
\[
4h + 6 = 14 - 12(16 - 8h)
\]
2. Distribute \(-12\) on the right side:
\[
4h + 6 = 14 - 192 + 96h
\]
\[
4h + 6 = -178 + 96h
\]
3. Rearrange the equation to collect all terms involving \(h\) on one side and constant terms on the other side:
\[
4h - 96h = -178 - 6
\]
\[
-92h = -184
\]
4. Divide both sides by \(-92\):
\[
h = \frac{-184}{-92} = 2
\]
Since we obtained a specific numeric value for \(h\), there is exactly **one solution** to the equation.
Thus, the correct response is:
\[
\text{one solution}
\]
Answered by
GPT-4o mini
Let's carefully re-evaluate the equation \(4h + 6 = 14 - 12(16 - 8h)\) step by step.
1. Start with the original equation:
\[
4h + 6 = 14 - 12(16 - 8h)
\]
2. Distribute the \(-12\) on the right side:
\[
4h + 6 = 14 - 192 + 96h
\]
This simplifies to:
\[
4h + 6 = -178 + 96h
\]
3. Rearranging the equation to isolate \(h\):
\[
4h - 96h = -178 - 6
\]
\[
-92h = -184
\]
4. Divide both sides by \(-92\):
\[
h = \frac{-184}{-92} = 2
\]
Now let’s ensure there are no mistakes during the simplification:
### Check the Steps:
- Recheck the distribution:
- \( -12(16) = -192 \)
- \( -12(-8h) = +96h \)
The right-hand side indeed simplifies to \( -178 + 96h \).
- Rearranging gives you:
- \( 4h - 96h = -178 - 6 \)
- Leads to \(-92h = -184\), which confirms \( h = 2 \).
### Conclusion:
Thus, the equation has exactly **one solution** \(h = 2\).
If you believe my answer is still incorrect, please provide any specific detail or clarification on the method or concept you would like me to review.
1. Start with the original equation:
\[
4h + 6 = 14 - 12(16 - 8h)
\]
2. Distribute the \(-12\) on the right side:
\[
4h + 6 = 14 - 192 + 96h
\]
This simplifies to:
\[
4h + 6 = -178 + 96h
\]
3. Rearranging the equation to isolate \(h\):
\[
4h - 96h = -178 - 6
\]
\[
-92h = -184
\]
4. Divide both sides by \(-92\):
\[
h = \frac{-184}{-92} = 2
\]
Now let’s ensure there are no mistakes during the simplification:
### Check the Steps:
- Recheck the distribution:
- \( -12(16) = -192 \)
- \( -12(-8h) = +96h \)
The right-hand side indeed simplifies to \( -178 + 96h \).
- Rearranging gives you:
- \( 4h - 96h = -178 - 6 \)
- Leads to \(-92h = -184\), which confirms \( h = 2 \).
### Conclusion:
Thus, the equation has exactly **one solution** \(h = 2\).
If you believe my answer is still incorrect, please provide any specific detail or clarification on the method or concept you would like me to review.
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