Determine which of the following statements is/are true:

Question

#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

1
1

2 and 3
2 and 3

1 and 2

Answer 1

Let's evaluate each statement one by one.

Statement #1: \( 3 \cdot 20 = 4(6 + 9) \)

Calculating each side:

• Left Side: \[ 3 \cdot 20 = 60 \]
• Right Side: \[ 4(6 + 9) = 4 \cdot 15 = 60 \]

Both sides are equal, so Statement #1 is true.

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Statement #2: \( 4 \cdot 12 = 6 \cdot (8 \cdot 2) 12 + 6 \cdot 64 - \sqrt{64} = 2 \cdot 4 \cdot 8 \)

Let's break it down:

• Left Side: \[ 4 \cdot 12 = 48 \]

Now check the right side:

• Evaluate \( 6 \cdot (8 \cdot 2) \): \[ 8 \cdot 2 = 16 \] \[ 6 \cdot 16 = 96 \]

• Now compute \( 6 \cdot 64 \): \[ 6 \cdot 64 = 384 \]

• Now compute \( \sqrt{64} \): \[ \sqrt{64} = 8 \]

Combine these: \[ 96 + 384 - 8 = 472 \]

Now check \( 2 \cdot 4 \cdot 8 \): \[ 2 \cdot 4 = 8 \] \[ 8 \cdot 8 = 64 \]

The right side should equal 64, but it actually came out to 472. Therefore, Statement #2 is false.

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Statement #3: \( 4 \cdot 18 - 4 \cdot 6 = 42 + 4 \cdot 8 \)

Calculating each side:

• Left Side: \[ 4 \cdot 18 - 4 \cdot 6 = 72 - 24 = 48 \]

• Right Side: \[ 42 + 4 \cdot 8 = 42 + 32 = 74 \]

Since the left side (48) is not equal to the right side (74), Statement #3 is false.

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Thus, the true statements gathered is only Statement #1.

Final answer: 1

Answer 2

Which expression is A, in order to have a true equation A=5−11/3?(1 point)
Responses

A=123−13
1 Start Fraction 2 over 3 End Fraction minus Start Fraction 1 over 3 End Fraction

A=153+13
Start Fraction 15 over 3 End Fraction plus Start Fraction 1 over 3 End Fraction

A=−123+13
negative 1 Start Fraction 2 over 3 End Fraction plus Start Fraction 1 over 3 End Fraction

A=123+13

Answer 3

To determine which expression \( A \) corresponds to the equation \( A = 5 - \frac{11}{3} \), let's first calculate \( 5 - \frac{11}{3} \).

We can start by converting \( 5 \) into a fraction with a denominator of \( 3 \):

\[ 5 = \frac{15}{3} \]

Now, we can substitute this back into the equation:

\[ A = \frac{15}{3} - \frac{11}{3} = \frac{15 - 11}{3} = \frac{4}{3} \]

Now we will evaluate each of the provided responses to see which one equals \( \frac{4}{3} \).

1. Option A: \( \frac{12}{3} - \frac{1}{3} \)

   \[ \frac{12}{3} - \frac{1}{3} = \frac{12 - 1}{3} = \frac{11}{3} \quad (\text{Not } \frac{4}{3}) \]

2. Option B: \( \frac{15}{3} + \frac{1}{3} \)

   \[ \frac{15}{3} + \frac{1}{3} = \frac{15 + 1}{3} = \frac{16}{3} \quad (\text{Not } \frac{4}{3}) \]

3. Option C: \( -\frac{12}{3} + \frac{1}{3} \)

   \[ -\frac{12}{3} + \frac{1}{3} = -4 + \frac{1}{3} = -\frac{12}{3}+\frac{1}{3} = -\frac{12 - 1}{3} = -\frac{11}{3} \quad (\text{Not } \frac{4}{3}) \]

4. Option D: \( \frac{12}{3} + \frac{1}{3} \)

   \[ \frac{12}{3} + \frac{1}{3} = \frac{12 + 1}{3} = \frac{13}{3} \quad (\text{Not } \frac{4}{3}) \]

After evaluating all the options, none of them yield \( \frac{4}{3} \).

Thus, it seems there might be a misalignment with the responses versus the calculated value of \( A \). Would you like to double-check the options or calculations provided?

Answer 4

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-or-equal-to

=
equals

<

Answer 5

Let's evaluate the expressions on both sides of the inequality:

Left Side:

The expression is \(-6 + 3 \cdot 5\).

Calculating this: \[ 3 \cdot 5 = 15 \] So, \[ -6 + 15 = 9 \]

Right Side:

The expression is \(16 - \sqrt{16} + 32 \div 8\).

Calculating it step by step:

1. \( \sqrt{16} = 4\)
2. \( 32 \div 8 = 4\)
3. Now evaluate the whole expression: \[ 16 - 4 + 4 = 16 - 4 + 4 \] This simplifies to: \[ 16 - 4 + 4 = 16 \]

Now, Compare Both Sides:

• Left Side: \(9\)
• Right Side: \(16\)

Now we need to find the operator that can make the statement true.

Options:

1. > (greater than): \(9 > 16\) is false.
2. ≤ (less than or equal to): \(9 ≤ 16\) is true.
3. = (equals): \(9 = 16\) is false.
4. < (less than): \(9 < 16\) is true.

Since both \(9 ≤ 16\) and \(9 < 16\) are true, the operator that can definitely be inserted to make the statement true is:

≤ (less-than-or-equal-to)

Thus the final answer is: ≤

Answer 6

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

22
22

23
23

23.5

Answer 7

We start with the inequality:

\[ 3r - 47.5 > 21.5 \]

To solve for \( r \), let's first isolate the term involving \( r \):

1. Add \( 47.5 \) to both sides: \[ 3r > 21.5 + 47.5 \] \[ 3r > 69 \]

2. Divide both sides by \( 3 \): \[ r > \frac{69}{3} \] \[ r > 23 \]

This means that \( r \) must be greater than \( 23 \) for the inequality \( 3r - 47.5 > 21.5 \) to be a true statement.

Now let's check the provided options:

1. 22.5: This is not greater than \( 23 \) (False)
2. 22: This is not greater than \( 23 \) (False)
3. 23: This is not greater than \( 23 \) (False)
4. 23.5: This is greater than \( 23 \) (True)

Conclusion:

The value of \( r \) that makes the inequality \( 3r - 47.5 > 21.5 \) a true statement is 23.5.