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
(1 point)
Responses
1 and 3
1 and 3
2 and 3
2 and 3
1 and 2
1 and 2
1
#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
(1 point)
Responses
1 and 3
1 and 3
2 and 3
2 and 3
1 and 2
1 and 2
1
Answers
Answer
Which expression is A, in order to have a true equation A=5−113 ?(1 point)
Responses
A=−123+13
negative 1 Start Fraction 2 over 3 End Fraction plus Start Fraction 1 over 3 End Fraction
A=123+13
negative 1 Start Fraction 2 over 3 End Fraction minus Start Fraction 1 over 3 End Fraction
A=123−13
1 Start Fraction 2 over 3 End Fraction minus Start Fraction 1 over 3 End Fraction
A=153+13
Responses
A=−123+13
negative 1 Start Fraction 2 over 3 End Fraction plus Start Fraction 1 over 3 End Fraction
A=123+13
negative 1 Start Fraction 2 over 3 End Fraction minus Start Fraction 1 over 3 End Fraction
A=123−13
1 Start Fraction 2 over 3 End Fraction minus Start Fraction 1 over 3 End Fraction
A=153+13
Answer
Consider the following expressions:
−6+3⋅5 ______ 16−−√+32÷8 .
Which operator can be inserted to make the statement true?
(1 point)
Responses
>
greater than
<
less than
≤
less-than-or-equal-to
=
−6+3⋅5 ______ 16−−√+32÷8 .
Which operator can be inserted to make the statement true?
(1 point)
Responses
>
greater than
<
less than
≤
less-than-or-equal-to
=
Answer
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? (1 point)
Responses
22.5
22.5
23.5
23.5
22
22
23
Responses
22.5
22.5
23.5
23.5
22
22
23
Answer
Which of the following values for x makes the inequality 4x+1−−−−−√ ≤5.83 true?(1 point)
Responses
8
8
10
10
9
9
9.5
Responses
8
8
10
10
9
9
9.5
Answered by
GPT-4 Turbo
Let's evaluate whether each statement is true:
#1: 3⋅20 = 4(6+9)
3⋅20 = 60
4(6+9) = 4⋅15 = 60
Both sides are equal, so statement #1 is true.
#2: 4×12 = 6×(8×2) 12+6⋅64−−√ = 2⋅4⋅8
This statement seems to have a formatting problem. Let's break it down step-by-step.
First part:
4×12 = 48
Second part:
6×(8×2) = 6×16 = 96
Third part:
12 + 6⋅64√
There appears to be a typo or error in the expression. If the intention is to take the square root of 64, it should be written as 12 + 6√64, which simplifies to:
12 + 6⋅8 = 12 + 48 = 60
Fourth part:
2⋅4⋅8 = 64
It's unclear how these components are meant to relate to each other since the formatting is off. However, from the separated calculations, we can see that they don't seem to match accordingly. Therefore, statement #2 is unclear or potentially false with the given information.
#3: 4⋅18−4⋅6 = 42+4⋅8
4⋅18−4⋅6 = 72−24 = 48
42+4⋅8 = 42+32 = 74
Both sides are not equal, so statement #3 is false.
Given these evaluations, and assuming that the malformed equation in statement #2 is incorrect, the correct response is:
1
#1: 3⋅20 = 4(6+9)
3⋅20 = 60
4(6+9) = 4⋅15 = 60
Both sides are equal, so statement #1 is true.
#2: 4×12 = 6×(8×2) 12+6⋅64−−√ = 2⋅4⋅8
This statement seems to have a formatting problem. Let's break it down step-by-step.
First part:
4×12 = 48
Second part:
6×(8×2) = 6×16 = 96
Third part:
12 + 6⋅64√
There appears to be a typo or error in the expression. If the intention is to take the square root of 64, it should be written as 12 + 6√64, which simplifies to:
12 + 6⋅8 = 12 + 48 = 60
Fourth part:
2⋅4⋅8 = 64
It's unclear how these components are meant to relate to each other since the formatting is off. However, from the separated calculations, we can see that they don't seem to match accordingly. Therefore, statement #2 is unclear or potentially false with the given information.
#3: 4⋅18−4⋅6 = 42+4⋅8
4⋅18−4⋅6 = 72−24 = 48
42+4⋅8 = 42+32 = 74
Both sides are not equal, so statement #3 is false.
Given these evaluations, and assuming that the malformed equation in statement #2 is incorrect, the correct response is:
1
Answered by
GPT-4 Turbo
Let's first clarify the value we're trying to match:
A = 5 - 1 1/3
This can be seen as 5 minus 1 and 1/3. To simplify, we convert 5 to a fractional form with a denominator of 3:
5 = 15/3 (since 5 = 15/3 * 1)
Now the expression becomes:
A = 15/3 - 1 1/3
To subtract 1 1/3 from 15/3, we first need to convert 1 1/3 to an improper fraction:
1 1/3 = (1*3 + 1)/3 = 4/3
Now we can subtract:
A = 15/3 - 4/3
A = (15-4)/3
A = 11/3
So, in terms of the responses provided:
A = −12/3 + 1/3
This simplifies to: A = −11/3. This is the opposite of what we want, as we're looking for 11/3.
A = 12/3 + 1/3
12/3 = 4, so A = 4 + 1/3, which is 4 1/3, or as an improper fraction, it's (4*3 + 1)/3 = 13/3. This is not the correct answer.
A = 12/3 − 1/3
Again, 12/3 = 4, so A = 4 - 1/3, which is 3 2/3, or as an improper fraction, it's (3*3 + 2)/3 = 11/3. This is the correct answer.
A = 15/3 + 1/3
15/3 is 5, so A = 5 + 1/3. This is 5 1/3, or as an improper fraction, it's (5*3 + 1)/3 = 16/3. This is not the correct answer.
Thus, the correct expression for A to have a true equation A = 5 - 1 1/3 is:
A = 12/3 − 1/3 (1 and 2/3 minus 1/3)
A = 5 - 1 1/3
This can be seen as 5 minus 1 and 1/3. To simplify, we convert 5 to a fractional form with a denominator of 3:
5 = 15/3 (since 5 = 15/3 * 1)
Now the expression becomes:
A = 15/3 - 1 1/3
To subtract 1 1/3 from 15/3, we first need to convert 1 1/3 to an improper fraction:
1 1/3 = (1*3 + 1)/3 = 4/3
Now we can subtract:
A = 15/3 - 4/3
A = (15-4)/3
A = 11/3
So, in terms of the responses provided:
A = −12/3 + 1/3
This simplifies to: A = −11/3. This is the opposite of what we want, as we're looking for 11/3.
A = 12/3 + 1/3
12/3 = 4, so A = 4 + 1/3, which is 4 1/3, or as an improper fraction, it's (4*3 + 1)/3 = 13/3. This is not the correct answer.
A = 12/3 − 1/3
Again, 12/3 = 4, so A = 4 - 1/3, which is 3 2/3, or as an improper fraction, it's (3*3 + 2)/3 = 11/3. This is the correct answer.
A = 15/3 + 1/3
15/3 is 5, so A = 5 + 1/3. This is 5 1/3, or as an improper fraction, it's (5*3 + 1)/3 = 16/3. This is not the correct answer.
Thus, the correct expression for A to have a true equation A = 5 - 1 1/3 is:
A = 12/3 − 1/3 (1 and 2/3 minus 1/3)
Answered by
GPT-4 Turbo
Let's first calculate the value of each expression:
The left-side expression:
−6 + 3⋅5
= −6 + 15
= 9
The right-side expression:
√16 + 32÷8
First, we calculate the square root and division:
√16 = 4 (since 4^2 = 16)
32÷8 = 4 (since 32 divided by 8 equals 4)
Now, we add these results together:
4 + 4 = 8
Now we compare both expressions:
Left-side expression (9) __ Right-side expression (8)
To determine the operator:
9 __ 8
Since 9 is greater than 8, the correct operator to make the statement true is:
>
So the complete true statement would be:
−6 + 3⋅5 > √16 + 32÷8
The left-side expression:
−6 + 3⋅5
= −6 + 15
= 9
The right-side expression:
√16 + 32÷8
First, we calculate the square root and division:
√16 = 4 (since 4^2 = 16)
32÷8 = 4 (since 32 divided by 8 equals 4)
Now, we add these results together:
4 + 4 = 8
Now we compare both expressions:
Left-side expression (9) __ Right-side expression (8)
To determine the operator:
9 __ 8
Since 9 is greater than 8, the correct operator to make the statement true is:
>
So the complete true statement would be:
−6 + 3⋅5 > √16 + 32÷8
Answered by
GPT-4 Turbo
First, let's verify that the equation 3r−47.5=21.5 is true when r=23.
3r - 47.5 = 21.5
3(23) - 47.5 = 21.5
69 - 47.5 = 21.5
21.5 = 21.5
The equation is indeed true when r is 23. Now let's find the value of r that makes the inequality 3r−47.5>21.5 true. We are looking for a value of r that when plugged into the inequality, gives a result greater than 21.5.
Using the same equation, but replacing the equal sign with a greater-than sign:
3r - 47.5 > 21.5
Let's solve for r:
3r > 21.5 + 47.5
3r > 69
Now, divide both sides by 3:
r > 69 / 3
r > 23
So r has to be greater than 23 for the inequality to be true. Out of the given options: 22.5, 23.5, 22, and 23, the value that is greater than 23 and thus makes the inequality true is:
23.5
Therefore, the correct response is 23.5.
3r - 47.5 = 21.5
3(23) - 47.5 = 21.5
69 - 47.5 = 21.5
21.5 = 21.5
The equation is indeed true when r is 23. Now let's find the value of r that makes the inequality 3r−47.5>21.5 true. We are looking for a value of r that when plugged into the inequality, gives a result greater than 21.5.
Using the same equation, but replacing the equal sign with a greater-than sign:
3r - 47.5 > 21.5
Let's solve for r:
3r > 21.5 + 47.5
3r > 69
Now, divide both sides by 3:
r > 69 / 3
r > 23
So r has to be greater than 23 for the inequality to be true. Out of the given options: 22.5, 23.5, 22, and 23, the value that is greater than 23 and thus makes the inequality true is:
23.5
Therefore, the correct response is 23.5.
Answered by
GPT-4 Turbo
To find which value of x makes the inequality \( \sqrt{4x + 1} \leq 5.83 \) true, we first need to solve the inequality for x.
From the given inequality, we can square both sides, as squaring is a valid operation when both sides are positive. Since square roots are always positive (or zero) and 5.83 is a positive number, we won't run into complications due to negative numbers.
Squaring both sides gives us:
\( (\sqrt{4x + 1})^2 \leq (5.83)^2 \)
\( 4x + 1 \leq 5.83^2 \)
Now calculate 5.83 squared:
\( 4x + 1 \leq 34.0089 \)
Subtract 1 from both sides to isolate the term with x:
\( 4x \leq 33.0089 \)
Now, divide both sides by 4:
\( x \leq \frac{33.0089}{4} \)
\( x \leq 8.252225 \)
This result means that x must be less than or equal to approximately 8.252225 for the inequality to be true.
Now we must review our response options:
- 8
- 10
- 9
- 9.5
Out of these options, the value 8 is the only one that is less than or equal to our calculated value of approximately 8.252225, so x = 8 makes the inequality \( \sqrt{4x + 1} \leq 5.83 \) true.
Thus, the correct response is 8.
From the given inequality, we can square both sides, as squaring is a valid operation when both sides are positive. Since square roots are always positive (or zero) and 5.83 is a positive number, we won't run into complications due to negative numbers.
Squaring both sides gives us:
\( (\sqrt{4x + 1})^2 \leq (5.83)^2 \)
\( 4x + 1 \leq 5.83^2 \)
Now calculate 5.83 squared:
\( 4x + 1 \leq 34.0089 \)
Subtract 1 from both sides to isolate the term with x:
\( 4x \leq 33.0089 \)
Now, divide both sides by 4:
\( x \leq \frac{33.0089}{4} \)
\( x \leq 8.252225 \)
This result means that x must be less than or equal to approximately 8.252225 for the inequality to be true.
Now we must review our response options:
- 8
- 10
- 9
- 9.5
Out of these options, the value 8 is the only one that is less than or equal to our calculated value of approximately 8.252225, so x = 8 makes the inequality \( \sqrt{4x + 1} \leq 5.83 \) true.
Thus, the correct response is 8.