Question

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

<

Answer 1

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

Answer 2

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

Answer 3

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

Answer 4

pick one of the options a gave you, give me the closest answer

Answer 5

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

Answer 6

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)

Answer 7

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**

Answer 8

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:

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

Answer 9

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: \(\frac{5}{6} \leq x < \frac{7}{3}\).

Answer 10

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.

Answer 11

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**.