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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

9 answers

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.

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

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.

Consider the following expressions:
−6+3⋅5 ______ 16−−√+32÷8 .
Which operator can be inserted to make the statement true?
Responses:

≤

=

<

>

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).

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

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.

Which of the following values for x makes the inequality 4x+1−−−−−√ ≤5.83 true?(1 point)
Responses:
9
8
9.5
10

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.