Ask a New Question
Menu

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 2
1 and 2

1 and 3
1 and 3

2 and 3
2 and 3

1.

11 answers

Let's evaluate each statement one by one to determine which are true.

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

Calculating each side:

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

So, both sides are equal: \( 60 = 60 \). Thus, statement #1 is true.

#2: \( 4 \cdot 12 = 6 \cdot (8 \cdot 2) \)

Calculating each side:

  • Left Side: \( 4 \cdot 12 = 48 \)
  • Right Side: \[ 6 \cdot (8 \cdot 2) = 6 \cdot 16 = 96 \]

Since \( 48 \neq 96 \), statement #2 is false.

#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 \( 48 \neq 74 \), statement #3 is false.

Now let's summarize the results:

  • Statement #1 is true.
  • Statement #2 is false.
  • Statement #3 is false.

Therefore, the only true statement is #1.

The correct response is: 1 and 2

Which expression is A, in order to have a true equation A=5−113 ?(1 point)
Responses

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
negative 1 Start Fraction 2 over 3 End Fraction minus Start Fraction 1 over 3 End Fraction

A=123−13

To determine which expression \( A \) makes the equation \( A = 5 - \frac{1}{3} \) true, we first need to evaluate the right side of the equation.

Calculating \( 5 - \frac{1}{3} \)

  1. Convert \( 5 \) into a fraction with a denominator of 3: \[ 5 = \frac{15}{3} \]

  2. Now, perform the subtraction: \[ 5 - \frac{1}{3} = \frac{15}{3} - \frac{1}{3} = \frac{15 - 1}{3} = \frac{14}{3} \]

So, \( A \) must equal \( \frac{14}{3} \).

Evaluating Each Response:

  1. \( A = \frac{15}{3} + \frac{1}{3} \): \[ A = \frac{15 + 1}{3} = \frac{16}{3} \quad (\text{Not equal to } \frac{14}{3}) \]

  2. \( A = -\frac{1}{2} + \frac{1}{3} \): To combine these fractions, we will need a common denominator:

    • The least common denominator of 2 and 3 is 6. \[ A = -\frac{3}{6} + \frac{2}{6} = -\frac{1}{6} \quad (\text{Not equal to } \frac{14}{3}) \]

  3. \( A = -\frac{1}{2} - \frac{1}{3} \): Again using a common denominator: \[ A = -\frac{3}{6} - \frac{2}{6} = -\frac{5}{6} \quad (\text{Not equal to } \frac{14}{3}) \]

  4. \( A = \frac{14}{3} \): This matches our calculated value of \( A \).

Conclusion:

The correct expression for \( A \) that makes the equation \( A = 5 - \frac{1}{3} \) true is: A = \frac{14}{3}.

It seems like the fourth response option you provided is missing from the list. Based on our evaluation, the only correct value that satisfies the equation \( 5 - \frac{1}{3} = \frac{14}{3} \) is \( A = \frac{14}{3} \). Could you please recheck your options?

Which expression is A, in order to have a true equation A=5−113?(1 point) Responses 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 negative 1 Start Fraction 2 over 3 End Fraction minus Start Fraction 1 over 3 End Fraction A=123−13

To determine which expression \( A \) satisfies the equation \( A = 5 - \frac{1}{3} \), we first calculate the right side of the equation.

Step 1: Calculate \( 5 - \frac{1}{3} \)

  1. Convert \( 5 \) into a fraction with a denominator of 3: \[ 5 = \frac{15}{3} \]

  2. Now perform the subtraction: \[ 5 - \frac{1}{3} = \frac{15}{3} - \frac{1}{3} = \frac{15 - 1}{3} = \frac{14}{3} \]

Step 2: Evaluate Each Response Option

Now, we need to see which of the provided expressions equals \( \frac{14}{3} \).

  1. \( A = \frac{15}{3} + \frac{1}{3} \): \[ A = \frac{15 + 1}{3} = \frac{16}{3} \quad (\text{Not equal to } \frac{14}{3}) \]

  2. \( A = -\frac{1}{2} + \frac{1}{3} \): To combine these fractions:

    • The least common denominator of 2 and 3 is 6. \[ A = -\frac{3}{6} + \frac{2}{6} = -\frac{1}{6} \quad (\text{Not equal to } \frac{14}{3}) \]

  3. \( A = -\frac{1}{2} - \frac{1}{3} \): Using a common denominator: \[ A = -\frac{3}{6} - \frac{2}{6} = -\frac{5}{6} \quad (\text{Not equal to } \frac{14}{3}) \]

  4. \( A = \frac{12}{3} - \frac{1}{3} \): \[ A = \frac{12 - 1}{3} = \frac{11}{3} \quad (\text{Not equal to } \frac{14}{3}) \]

Conclusion

None of the provided responses equal \( \frac{14}{3} \), which is the value we calculated for \( A \).

However, if we evaluate closely:

  • The fourth option \( A = \frac{12}{3} - \frac{1}{3} \) might have been misinterpreted originally, as it is indeed \( \frac{11}{3} \).
  • The remaining potential options either result in fractions that do not equal \( \frac{14}{3} \).

If \( A \) must equal \( \frac{14}{3} \), the expressions need to be checked for correctness against this result:

The correct expression \( A \) for \( A = \frac{14}{3} \) is not present among the options provided. Please verify if there might be any errors in the option descriptions or provide additional details if available.

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

=
equals

≤
less-than-or-equal-to

<

Let's evaluate both sides of the expression to determine what operator can be inserted to make the equation true.

Left Side: \( -6 + 3 \cdot 5 \)

Calculate step by step:

  1. \( 3 \cdot 5 = 15 \)
  2. Then, \( -6 + 15 = 9 \)

So, the left side evaluates to: \[ -6 + 3 \cdot 5 = 9 \]

Right Side: \( \sqrt{16} + 32 \div 8 \)

Calculate step by step:

  1. \( \sqrt{16} = 4 \)
  2. \( 32 \div 8 = 4 \)
  3. Then, \( 4 + 4 = 8 \)

So, the right side evaluates to: \[ \sqrt{16} + 32 \div 8 = 4 + 4 = 8 \]

Comparing the Two Sides

Now we have:

  • Left Side: \( 9 \)
  • Right Side: \( 8 \)

We need to analyze the relationship between the two sides:

  • \( 9 > 8 \)
  • Thus, the correct operator to make the statement true is \( > \) (greater than).

Conclusion

The operator that can be inserted to make the statement true is: > (greater than).

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

23
23

22
22

22.5
22.5

23.5

To determine which value of \( r \) satisfies the inequality \( 3r - 47.5 > 21.5 \), let's first simplify the inequality.

Step 1: Solve the Inequality

Starting from the inequality: \[ 3r - 47.5 > 21.5 \]

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

Now, divide both sides by 3 to isolate \( r \): \[ r > \frac{69}{3} \] \[ r > 23 \]

Step 2: Compare with the Given Options

Now, we check which of the provided options is greater than 23:

  1. 23: This is not greater than 23.
  2. 22: This is not greater than 23.
  3. 22.5: This is not greater than 23.
  4. 23.5: This is greater than 23.

Conclusion

The value of \( r \) that satisfies the inequality \( 3r - 47.5 > 21.5 \) is: 23.5.

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

9.5
9.5

9
9

8
8

10

To determine which value of \( x \) satisfies the inequality \( \sqrt{4x + 1} \leq 5.83 \), we first need to solve the inequality.

Step 1: Solve the Inequality

Starting from the inequality: \[ \sqrt{4x + 1} \leq 5.83 \]

Step 2: Square Both Sides

To eliminate the square root, square both sides (valid since both sides are non-negative): \[ 4x + 1 \leq (5.83)^2 \]

Calculating \( (5.83)^2 \): \[ 5.83^2 = 33.9889 \]

Now substitute this value back into the inequality: \[ 4x + 1 \leq 33.9889 \]

Step 3: Isolate \( x \)

Subtract 1 from both sides: \[ 4x \leq 33.9889 - 1 \] \[ 4x \leq 32.9889 \]

Now, divide both sides by 4: \[ x \leq \frac{32.9889}{4} \] Calculating the division: \[ x \leq 8.247225 \]

Step 4: Compare with the Given Options

Now, compare \( x \leq 8.247225 \) with the provided options:

  1. 9.5: Not valid (9.5 > 8.247225)
  2. 9: Not valid (9 > 8.247225)
  3. 8: Valid (8 ≤ 8.247225)
  4. 10: Not valid (10 > 8.247225)

Conclusion

The value of \( x \) that satisfies the inequality \( \sqrt{4x + 1} \leq 5.83 \) is: 8.