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Is the equation 13/4−3=1/4 true or false? Enter 1 if the statement is true. Enter 2 if the statement is false.(1 point)

9 answers

To determine if the equation \( \frac{13}{4} - 3 = \frac{1}{4} \) is true or false, let's simplify the left side:

  1. Convert \( 3 \) to a fraction with a denominator of \( 4 \): \[ 3 = \frac{12}{4} \]

  2. Substitute this into the equation: \[ \frac{13}{4} - \frac{12}{4} = \frac{1}{4} \]

  3. Now subtract the fractions: \[ \frac{13 - 12}{4} = \frac{1}{4} \] \[ \frac{1}{4} = \frac{1}{4} \]

Since both sides are equal, the equation is true.

Thus, the answer is \( 1 \).

Consider the following expressions: −34+6⋅216−−√ _____ 4(2+6)÷2−6 . Which operator ( > , < , or =) should be inserted into the blank space to make the statement true?​ (1 point)

()

To compare the two expressions in the statement \(-34 + 6 \cdot \sqrt{216} , __ , 4(2 + 6) \div 2 - 6\), we need to calculate both sides.

Left Side: \(-34 + 6 \cdot \sqrt{216}\)

  1. Calculate \(\sqrt{216}\): \[ \sqrt{216} = \sqrt{36 \times 6} = 6\sqrt{6} \approx 6 \times 2.45 \approx 14.7 \]
  2. Now compute \(6 \cdot \sqrt{216}\): \[ 6 \cdot \sqrt{216} \approx 6 \cdot 14.7 \approx 88.2 \]
  3. Now, compute \(-34 + 6 \cdot \sqrt{216}\): \[ -34 + 88.2 \approx 54.2 \]

Right Side: \(4(2 + 6) \div 2 - 6\)

  1. Calculate \(2 + 6\): \[ 2 + 6 = 8 \]
  2. Now compute \(4 \cdot 8 \div 2\): \[ 4 \cdot 8 = 32 \] \[ 32 \div 2 = 16 \]
  3. Finally, compute \(16 - 6\): \[ 16 - 6 = 10 \]

Comparison:

Now we compare the two results:

  • Left Side: \(\approx 54.2\)
  • Right Side: \(10\)

So we have: \[ 54.2 , __ , 10 \]

Since \(54.2 > 10\), the operator that should be inserted is \(>\).

Thus, the answer is \(>\).

Consider this inequality: 5b−2>13 . Does the value of 3.5 for b make the inequality a true statement? Enter 1 if the answer is yes. Enter 2 if the answer is no. (1 point)

()

To determine if the value \( b = 3.5 \) makes the inequality \( 5b - 2 > 13 \) a true statement, we can substitute \( b \) into the inequality and simplify:

  1. Substitute \( b = 3.5 \): \[ 5(3.5) - 2 > 13 \]

  2. Calculate \( 5(3.5) \): \[ 5(3.5) = 17.5 \]

  3. Now substitute back into the inequality: \[ 17.5 - 2 > 13 \]

  4. Simplify the left side: \[ 17.5 - 2 = 15.5 \]

  5. Now check the inequality: \[ 15.5 > 13 \]

Since this statement is true, the value of \( 3.5 \) for \( b \) makes the inequality a true statement.

Thus, the answer is \( 1 \).

Which value of x makes the equation x−−√+2=7 true: 20, 25, or 30?(1 point)

x=()

To solve the equation \(\sqrt{x} + 2 = 7\), we need to isolate \(\sqrt{x}\):

  1. Subtract 2 from both sides: \[ \sqrt{x} = 7 - 2 \] \[ \sqrt{x} = 5 \]

  2. Now, square both sides to eliminate the square root: \[ x = 5^2 \] \[ x = 25 \]

Now we check the given options: 20, 25, or 30.

The value that makes the equation true is: \[ \boxed{25} \]

So, the answer is \(x = 25\).

Consider the equation 2z−2−−−−−√=5 . Which of the following values for z makes the equation true: 11.5, 12.5, or 13.5?(1 point)

The value of z that makes the equation true is()

To solve the equation \( \sqrt{2z - 2} = 5 \), we will first isolate the square root and then square both sides to eliminate it.

  1. Square both sides of the equation: \[ (\sqrt{2z - 2})^2 = 5^2 \] \[ 2z - 2 = 25 \]

  2. Add 2 to both sides: \[ 2z = 25 + 2 \] \[ 2z = 27 \]

  3. Divide both sides by 2 to solve for \( z \): \[ z = \frac{27}{2} = 13.5 \]

Now we check the provided options: 11.5, 12.5, or 13.5.

The value that makes the equation true is: \[ \boxed{13.5} \]

So, the answer is \( z = 13.5 \).