Question
Consider the following expressions:
6−14⋅28−214 _______ 34+4⋅2−11 .
Which operator can be inserted to make the statement true?
(1 point)
Responses
>
greater than
≥
greater-than-or-equal-to
<
less than
=
6−14⋅28−214 _______ 34+4⋅2−11 .
Which operator can be inserted to make the statement true?
(1 point)
Responses
>
greater than
≥
greater-than-or-equal-to
<
less than
=
Answers
Answer
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
negative 12
−13
negative 13
−12.5
Responses
−13.5
negative 13.5
−12
negative 12
−13
negative 13
−12.5
Answer
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 plus 8 equals negative 4
4x+8=4
4 x plus 8 equals 4
4x−8=4
Responses
4x−8=−4
4 x minus 8 equals negative 4
4x+8=−4
4 x plus 8 equals negative 4
4x+8=4
4 x plus 8 equals 4
4x−8=4
Answer
Which set expresses the solution to the compound inequality −2≤2x+13−4<1?(1 point)
Responses
{x:−32≤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:3≤x<7}
left brace x colon 3 less-than-or-equal-to x less than 7 right brace
{x:52≤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:52≤x<7}
left brace x colon Start Fraction 5 over 2 End Fraction less-than-or-equal-to x less than 7 right brace
Responses
{x:−32≤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:3≤x<7}
left brace x colon 3 less-than-or-equal-to x less than 7 right brace
{x:52≤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:52≤x<7}
left brace x colon Start Fraction 5 over 2 End Fraction less-than-or-equal-to x less than 7 right brace
Answer
just pick one
Answer
What property of equality is displayed in step 3 in the solving of 9m+4−7m=5m+73?
Step 1:
9m+4−7m=5m+73
9m−7m+4=5m+73
Step 2:
2m+4=5m+73
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
Multiplication Property of Equality
Multiplication Property of Equality
Addition Property of Equality
Addition Property of Equality
Subtraction Property of Equality
Subtraction Property of Equality
Division Property of Equality
Step 1:
9m+4−7m=5m+73
9m−7m+4=5m+73
Step 2:
2m+4=5m+73
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
Multiplication Property of Equality
Multiplication Property of Equality
Addition Property of Equality
Addition Property of Equality
Subtraction Property of Equality
Subtraction Property of Equality
Division Property of Equality
Answered by
GPT-4o mini
To determine which operator can be inserted to make the statement true, we first need to evaluate both sides of the expression.
**Left Side:**
\( 6 - 14 \cdot 28 - 214 \)
Calculating:
1. \( 14 \cdot 28 = 392 \)
2. \( 6 - 392 - 214 = 6 - 392 - 214 \)
3. \( 6 - 392 = -386 \)
4. \( -386 - 214 = -600 \)
So, the left side equals \( -600 \).
**Right Side:**
\( 34 + 4 \cdot 2 - 11 \)
Calculating:
1. \( 4 \cdot 2 = 8 \)
2. \( 34 + 8 - 11 = 34 + 8 = 42 \)
3. \( 42 - 11 = 31 \)
So, the right side equals \( 31 \).
Now we compare the two results:
- Left Side: \( -600 \)
- Right Side: \( 31 \)
Now we can check the operators:
1. **Greater than (>)**: \( -600 > 31 \) (False)
2. **Greater than or equal to (≥)**: \( -600 ≥ 31 \) (False)
3. **Less than (<)**: \( -600 < 31 \) (True)
4. **Equal to (=)**: \( -600 = 31 \) (False)
The only operator that makes the statement true is **< (less than)**.
Thus, the correct response is:
< (less than)
**Left Side:**
\( 6 - 14 \cdot 28 - 214 \)
Calculating:
1. \( 14 \cdot 28 = 392 \)
2. \( 6 - 392 - 214 = 6 - 392 - 214 \)
3. \( 6 - 392 = -386 \)
4. \( -386 - 214 = -600 \)
So, the left side equals \( -600 \).
**Right Side:**
\( 34 + 4 \cdot 2 - 11 \)
Calculating:
1. \( 4 \cdot 2 = 8 \)
2. \( 34 + 8 - 11 = 34 + 8 = 42 \)
3. \( 42 - 11 = 31 \)
So, the right side equals \( 31 \).
Now we compare the two results:
- Left Side: \( -600 \)
- Right Side: \( 31 \)
Now we can check the operators:
1. **Greater than (>)**: \( -600 > 31 \) (False)
2. **Greater than or equal to (≥)**: \( -600 ≥ 31 \) (False)
3. **Less than (<)**: \( -600 < 31 \) (True)
4. **Equal to (=)**: \( -600 = 31 \) (False)
The only operator that makes the statement true is **< (less than)**.
Thus, the correct response is:
< (less than)
Answered by
GPT-4o mini
To solve the inequality \( -32.5 - 6b \geq 48.5 \), we'll first isolate \( b \).
1. Start by adding \( 32.5 \) to both sides:
\[
-6b \geq 48.5 + 32.5
\]
\[
-6b \geq 81
\]
2. Next, divide both sides by \(-6\). Remember that when you divide an inequality by a negative number, you need to reverse the inequality sign:
\[
b \leq \frac{81}{-6}
\]
\[
b \leq -13.5
\]
Now that we have the value for \( b \), we can check which of the provided options satisfy this inequality:
- **-13.5:** \( -13.5 \leq -13.5\) (True)
- **-12:** \( -12 \leq -13.5\) (False)
- **-13:** \( -13 \leq -13.5\) (False)
- **-12.5:** \( -12.5 \leq -13.5\) (False)
The only value that satisfies the inequality \( b \leq -13.5 \) is:
**−13.5 (negative 13.5)**.
1. Start by adding \( 32.5 \) to both sides:
\[
-6b \geq 48.5 + 32.5
\]
\[
-6b \geq 81
\]
2. Next, divide both sides by \(-6\). Remember that when you divide an inequality by a negative number, you need to reverse the inequality sign:
\[
b \leq \frac{81}{-6}
\]
\[
b \leq -13.5
\]
Now that we have the value for \( b \), we can check which of the provided options satisfy this inequality:
- **-13.5:** \( -13.5 \leq -13.5\) (True)
- **-12:** \( -12 \leq -13.5\) (False)
- **-13:** \( -13 \leq -13.5\) (False)
- **-12.5:** \( -12.5 \leq -13.5\) (False)
The only value that satisfies the inequality \( b \leq -13.5 \) is:
**−13.5 (negative 13.5)**.
Answered by
GPT-4o mini
To find which equation has the solution set \(\{-3\}\), we can substitute \(x = -3\) into each equation and see if the equation holds true.
1. **For \(4x - 8 = -4\)**:
\[
4(-3) - 8 = -4 \\
-12 - 8 = -4 \\
-20 \neq -4 \quad (\text{False})
\]
2. **For \(4x + 8 = -4\)**:
\[
4(-3) + 8 = -4 \\
-12 + 8 = -4 \\
-4 = -4 \quad (\text{True})
\]
3. **For \(4x + 8 = 4\)**:
\[
4(-3) + 8 = 4 \\
-12 + 8 = 4 \\
-4 \neq 4 \quad (\text{False})
\]
4. **For \(4x - 8 = 4\)**:
\[
4(-3) - 8 = 4 \\
-12 - 8 = 4 \\
-20 \neq 4 \quad (\text{False})
\]
After evaluating all equations, we can see that only the second equation, \(4x + 8 = -4\), has the solution \(x = -3\).
Thus, the correct response is:
**4x + 8 = -4 (4 x plus 8 equals negative 4)**.
1. **For \(4x - 8 = -4\)**:
\[
4(-3) - 8 = -4 \\
-12 - 8 = -4 \\
-20 \neq -4 \quad (\text{False})
\]
2. **For \(4x + 8 = -4\)**:
\[
4(-3) + 8 = -4 \\
-12 + 8 = -4 \\
-4 = -4 \quad (\text{True})
\]
3. **For \(4x + 8 = 4\)**:
\[
4(-3) + 8 = 4 \\
-12 + 8 = 4 \\
-4 \neq 4 \quad (\text{False})
\]
4. **For \(4x - 8 = 4\)**:
\[
4(-3) - 8 = 4 \\
-12 - 8 = 4 \\
-20 \neq 4 \quad (\text{False})
\]
After evaluating all equations, we can see that only the second equation, \(4x + 8 = -4\), has the solution \(x = -3\).
Thus, the correct response is:
**4x + 8 = -4 (4 x plus 8 equals negative 4)**.
Answered by
GPT-4o mini
Let's solve the compound inequality \( -2 \leq 2x + 13 - 4 < 1 \).
First, simplify the middle part of the inequality:
\[
-2 \leq 2x + 9 < 1
\]
Now, we'll break this down into two separate inequalities:
1. **First inequality**:
\[
-2 \leq 2x + 9
\]
Subtract \(9\) from both sides:
\[
-2 - 9 \leq 2x \\
-11 \leq 2x
\]
Now, divide by \(2\):
\[
-\frac{11}{2} \leq x
\]
Which can also be written as:
\[
x \geq -\frac{11}{2}
\]
2. **Second inequality**:
\[
2x + 9 < 1
\]
Subtract \(9\) from both sides:
\[
2x < 1 - 9 \\
2x < -8
\]
Now, divide by \(2\):
\[
x < -4
\]
Now we have two parts of the solution:
1. \( x \geq -\frac{11}{2} \)
2. \( x < -4 \)
The compound inequality can be combined:
\[
-\frac{11}{2} \leq x < -4
\]
Now let's express \(-\frac{11}{2}\) as a decimal:
\[
-\frac{11}{2} = -5.5
\]
Thus, we have:
\[
-5.5 \leq x < -4
\]
Finally, we need to compare this to the response options:
1. **\(\{x: -\frac{3}{2} \leq x < 1\}\)** (False)
2. **\(\{x: 3 \leq x < 7\}\)** (False)
3. **\(\{x: \frac{5}{2} \leq x < 1\}\)** (False)
4. **\(\{x: \frac{5}{2} \leq x < 7\}\)** (False)
None of the provided responses match the solution we derived. It seems that the responses do not include the correct interval.
So, the correct solution set would be:
\[
\{x: -\frac{11}{2} \leq x < -4\}
\]
But since this is not listed among the options, it appears there is no valid response given the original options. Please double-check the responses or provide additional context.
First, simplify the middle part of the inequality:
\[
-2 \leq 2x + 9 < 1
\]
Now, we'll break this down into two separate inequalities:
1. **First inequality**:
\[
-2 \leq 2x + 9
\]
Subtract \(9\) from both sides:
\[
-2 - 9 \leq 2x \\
-11 \leq 2x
\]
Now, divide by \(2\):
\[
-\frac{11}{2} \leq x
\]
Which can also be written as:
\[
x \geq -\frac{11}{2}
\]
2. **Second inequality**:
\[
2x + 9 < 1
\]
Subtract \(9\) from both sides:
\[
2x < 1 - 9 \\
2x < -8
\]
Now, divide by \(2\):
\[
x < -4
\]
Now we have two parts of the solution:
1. \( x \geq -\frac{11}{2} \)
2. \( x < -4 \)
The compound inequality can be combined:
\[
-\frac{11}{2} \leq x < -4
\]
Now let's express \(-\frac{11}{2}\) as a decimal:
\[
-\frac{11}{2} = -5.5
\]
Thus, we have:
\[
-5.5 \leq x < -4
\]
Finally, we need to compare this to the response options:
1. **\(\{x: -\frac{3}{2} \leq x < 1\}\)** (False)
2. **\(\{x: 3 \leq x < 7\}\)** (False)
3. **\(\{x: \frac{5}{2} \leq x < 1\}\)** (False)
4. **\(\{x: \frac{5}{2} \leq x < 7\}\)** (False)
None of the provided responses match the solution we derived. It seems that the responses do not include the correct interval.
So, the correct solution set would be:
\[
\{x: -\frac{11}{2} \leq x < -4\}
\]
But since this is not listed among the options, it appears there is no valid response given the original options. Please double-check the responses or provide additional context.
Answered by
GPT-4o mini
Given the options and since none of them exactly match the solution we derived \( -\frac{11}{2} \leq x < -4 \), I'll choose the option that is the closest based on the typical structure of such problems.
However, from the provided options, I notice none refer to negative values less than -4, but I'll select:
**\(\{x: -\frac{3}{2} \leq x < 1\}\)**
This is not a correct match, but it's the only set provided, and the others definitely don't fit.
However, from the provided options, I notice none refer to negative values less than -4, but I'll select:
**\(\{x: -\frac{3}{2} \leq x < 1\}\)**
This is not a correct match, but it's the only set provided, and the others definitely don't fit.
Answered by
GPT-4o mini
In Step 3, the transformation involved is:
\[
6m + 12 - 12 = 5m + 7 - 12
\]
Here, \(12\) is being subtracted from both sides of the equation.
This demonstrates the **Subtraction Property of Equality**, which states that if you subtract the same value from both sides of an equation, the equality is maintained.
Therefore, the correct response is:
**Subtraction Property of Equality**.
\[
6m + 12 - 12 = 5m + 7 - 12
\]
Here, \(12\) is being subtracted from both sides of the equation.
This demonstrates the **Subtraction Property of Equality**, which states that if you subtract the same value from both sides of an equation, the equality is maintained.
Therefore, the correct response is:
**Subtraction Property of Equality**.
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