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6−1/4⋅28−21/4

_______ 3/4+4⋅2−11
.

Which operator can be inserted to make the statement true?

(1 point)
Responses

>
greater than

≥
greater-than-or-equal-to

=
equals

<

11 answers

To determine which operator can be inserted to make the statement true, we first need to evaluate both sides of the equation.

The left side:

\(6 - \frac{1}{4} \cdot 28 - \frac{21}{4}\)

  1. Calculate \(\frac{1}{4} \cdot 28\): \(\frac{1}{4} \cdot 28 = 7\)

  2. Now substitute this back into the expression: \(6 - 7 - \frac{21}{4}\)

  3. Convert \(6\) to a fraction with a denominator of \(4\): \(6 = \frac{24}{4}\)

  4. Now we have: \(\frac{24}{4} - 7 - \frac{21}{4} = \frac{24}{4} - \frac{28}{4} = \frac{-4}{4} = -1\)

So the left side evaluates to \(-1\).

Now for the right side:

\(\frac{3}{4} + 4 \cdot 2 - 11\)

  1. Calculate \(4 \cdot 2\): \(4 \cdot 2 = 8\)

  2. Now substitute this back into the expression: \(\frac{3}{4} + 8 - 11\)

  3. Convert \(8\) and \(11\) to fractions with a denominator of \(4\): \(8 = \frac{32}{4}\) and \(11 = \frac{44}{4}\)

  4. Now we have: \(\frac{3}{4} + \frac{32}{4} - \frac{44}{4} = \frac{3 + 32 - 44}{4} = \frac{-9}{4}\)

So the right side evaluates to \(-\frac{9}{4}\).

Now we compare the two sides:

Left side: \(-1 = \frac{-4}{4}\)
Right side: \(-\frac{9}{4}\)

Now we can summarize:

\(-1 < -\frac{9}{4}\)

Therefore, we choose the operator:

\(<\)

So the answer is:

< (less than)

Which of the following values for b makes the inequality −32.5−6b≥48.5
true?(1 point)
Responses

−13.5
negative 13.5

−12.5
negative 12.5

−13
negative 13

−12

To solve the inequality \(-32.5 - 6b \geq 48.5\), we will isolate \(b\).

  1. Add 32.5 to both sides: \[ -6b \geq 48.5 + 32.5 \] \[ -6b \geq 81 \]

  2. Divide by -6: Remember that dividing or multiplying by a negative number reverses the inequality. \[ b \leq \frac{81}{-6} \] \[ b \leq -13.5 \]

Now, we need to select a value for \(b\) from the given options that satisfies this inequality:

  • \(-13.5 \) ⇒ True (it is equal to -13.5 which is allowed as per the inequality)
  • \(-12.5 \) ⇒ False (it is greater than -13.5)
  • \(-13 \) ⇒ False (it is greater than -13.5)
  • \(-12 \) ⇒ False (it is greater than -13.5)

Thus, the only value that makes the inequality true is:

−13.5

Which of the following equations has the solution set {−3}
?(1 point)
Responses

4x−8=−4
4 x minus 8 equals negative 4

4x−8=4
4 x minus 8 equals 4

4x+8=−4
4 x plus 8 equals negative 4

4x+8=4
4 x plus 8 equals 4

To determine which equation has the solution set \({-3}\), we will substitute \(-3\) into each equation and see which one results in a true statement.

  1. For the equation \(4x - 8 = -4\): \[ 4(-3) - 8 = -4 \ -12 - 8 = -4 \ -20 \neq -4 \quad \text{(False)} \]

  2. For the equation \(4x - 8 = 4\): \[ 4(-3) - 8 = 4 \ -12 - 8 = 4 \ -20 \neq 4 \quad \text{(False)} \]

  3. For the equation \(4x + 8 = -4\): \[ 4(-3) + 8 = -4 \ -12 + 8 = -4 \ -4 = -4 \quad \text{(True)} \]

  4. For the equation \(4x + 8 = 4\): \[ 4(-3) + 8 = 4 \ -12 + 8 = 4 \ -4 \neq 4 \quad \text{(False)} \]

Thus, the only equation that yields a true statement when \(x = -3\) is:

\(4x + 8 = -4\).

Which set expresses the solution to the compound inequality −2≤2x+1/3−4<1
?(1 point)
Responses

{x:−3/2≤x<1}
left brace x colon Start Fraction negative 3 over 2 End Fraction less-than-or-equal-to x less than 1 right brace

{x:5/2≤x<1}
left brace x colon Start Fraction 5 over 2 End Fraction less-than-or-equal-to x less than 1 right brace

{x:5/2≤x<7}
left brace x colon Start Fraction 5 over 2 End Fraction less-than-or-equal-to x less than 7 right brace

{x:3≤x<7}
left brace x colon 3 less-than-or-equal-to x less than 7 right brace

To solve the compound inequality \(-2 \leq 2x + \frac{1}{3} - 4 < 1\), we can break it into two separate inequalities and solve them one at a time.

  1. Solving the left part: \[ -2 \leq 2x + \frac{1}{3} - 4 \] First, simplify \( \frac{1}{3} - 4\): \[ -2 \leq 2x - \frac{11}{3} \]

    Add \(\frac{11}{3}\) to both sides: \[ -2 + \frac{11}{3} \leq 2x \]

    Convert \(-2\) to a fraction with a denominator of \(3\): \[ -2 = \frac{-6}{3} \]

    Now: \[ \frac{-6}{3} + \frac{11}{3} \leq 2x \ \frac{5}{3} \leq 2x \]

    Divide by \(2\): \[ \frac{5}{3} \cdot \frac{1}{2} \leq x \ \frac{5}{6} \leq x \]

  2. Solving the right part: \[ 2x + \frac{1}{3} - 4 < 1 \] Again, simplify \( \frac{1}{3} - 4\): \[ 2x - \frac{11}{3} < 1 \]

    Add \(\frac{11}{3}\) to both sides: \[ 2x < 1 + \frac{11}{3} \]

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

    Now: \[ 2x < \frac{3}{3} + \frac{11}{3} \ 2x < \frac{14}{3} \]

    Divide by \(2\): \[ x < \frac{14}{3} \cdot \frac{1}{2} \ x < \frac{7}{3} \]

  3. Putting it all together: From the left and right inequalities, we have: \[ \frac{5}{6} \leq x < \frac{7}{3} \]

Now, we check the options provided for their equivalence to the calculated range.

  • \({x : -\frac{3}{2} \leq x < 1}\) does not fit.
  • \({x : \frac{5}{2} \leq x < 1}\) does not fit.
  • \({x : \frac{5}{2} \leq x < 7}\) does not fit.
  • \({x : 3 \leq x < 7}\) does not fit.

None of the provided options match the calculated solution set of \(\frac{5}{6} \leq x < \frac{7}{3}\).

Given the question context, it appears that I may have made an error, or the correct option for the compound inequality may not be explicitly listed. To clarify:

  • The correct solubility indicates that we are seeking values in that defined range, and I'd recommend reconfirming the available choices or reviewing further context.

Overall, we deduced:

  • Final compound inequality: \(\frac{5}{6} \leq x < \frac{7}{3}\).
pick one of the options a gave you, give me the closest answer

Given your options, the closest answer to the calculated inequality \(\frac{5}{6} \leq x < \frac{7}{3}\) is:

\({x : -\frac{3}{2} \leq x < 1}\)

While this option does not match perfectly with \(\frac{5}{6} \approx 0.833\) and \( \frac{7}{3} \approx 2.33\), it is the closest of the provided choices based on the left endpoint being a lesser value than \(\frac{5}{6}\).

If we were to prioritize the closest range to \(\frac{5}{6}\) but less than \(1\), this option seems to be the most reasonable available choice.

What property of equality is displayed in step 3 in the solving of 9m+4−7m=5m+7/3
?

Step 1:

9m+4−7m=5m+7/3

9m−7m+4=5m+7/3

Step 2:

2m+4=5m+7/3

3(2m+4)=(5m+73)/3

Step 3:

6m+12=5m+7

6m+12−12=5m+7−12

Step 4:

6m=5m−5

6m−5m=5m−5m−5

m=−5

(1 point)
Responses

Addition Property of Equality
Addition Property of Equality

Multiplication Property of Equality
Multiplication Property of Equality

Subtraction Property of Equality
Subtraction Property of Equality

Division Property of Equality

In Step 3 of the solving process, the equation is:

\[ 6m + 12 - 12 = 5m + 7 - 12 \]

Here, both sides of the equation are being reduced by subtracting \(12\).

This demonstrates the Subtraction Property of Equality, which states that if you subtract the same number from both sides of an equation, the equality remains true.

Thus, the correct answer is:

Subtraction Property of Equality.