Asked by hi

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

<
less than

=
equals

Answers

Answered by hi
a,b,c,or d
Answered by hi
Which of the following values for b makes the inequality −32.5−6b≥48.5
true?(1 point)
Responses

−12.5
negative 12.5

−13
negative 13

−12
negative 12

−13.5
Answered by hi
Which of the following equations has the solution set {−3}
?(1 point)
Responses

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

4x−8=4
4 x minus 8 equals 4

4x+8=−4
4 x plus 8 equals negative 4

4x−8=−4
Answered by hi
Which set expresses the solution to the compound inequality −2≤2x+13−4<1
?(1 point)
Responses

{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

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

{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:52≤x<1}
Answered by hi
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

Division Property of Equality
Division Property of Equality

Multiplication Property of Equality
Multiplication Property of Equality

Addition Property of Equality
Addition Property of Equality

Subtraction 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 expressions on either side of the operator.

**Left-hand side evaluation:**

\[ 6 - 14 \cdot 28 - 214 \]

Calculating it step-by-step:

1. Calculate \( 14 \cdot 28 \):
\[ 14 \cdot 28 = 392 \]
2. Subtract \( 392 \) and \( 214 \) from \( 6 \):
\[ 6 - 392 - 214 = 6 - 392 - 214 \]
\[ = 6 - 606 \]
\[ = -600 \]

So, the left-hand side evaluates to \( -600 \).

**Right-hand side evaluation:**

\[ 34 + 4 \cdot 2 - 11 \]

Calculating it step-by-step:

1. Calculate \( 4 \cdot 2 \):
\[ 4 \cdot 2 = 8 \]
2. Add \( 34 \) and subtract \( 11 \):
\[ 34 + 8 - 11 = 34 + 8 = 42 - 11 \]
\[ = 31 \]

So, the right-hand side evaluates to \( 31 \).

**Comparing the results:**

The left-hand side is \( -600 \) and the right-hand side is \( 31 \).

Now we can determine which operator fits:

\[ -600 ? 31 \]

- **Greater than (>)**: \( -600 > 31 \) is false.
- **Less than (<)**: \( -600 < 31 \) is true.
- **Equals (=)**: \( -600 = 31 \) is false.
- **Greater than or equal (≥)**: \( -600 ≥ 31 \) is false.

The only true statement is:

\[ -600 < 31 \]

So, the correct operator to insert is:

**< (less than)**.
Answered by GPT-4o mini
Based on the evaluation we performed, the correct operator to insert is **b) < (less than)**.

Thus, the answer is:

**b)** <
Answered by GPT-4o mini
To determine which value for \( b \) makes the inequality \( -32.5 - 6b \geq 48.5 \) true, we first need to solve the inequality for \( b \).

1. Start with the inequality:
\[
-32.5 - 6b \geq 48.5
\]

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

3. Divide both sides by \( -6 \). Remember that when dividing by a negative number, we need to flip the inequality sign:
\[
b \leq \frac{81}{-6}
\]
\[
b \leq -13.5
\]

Now, let's evaluate the provided options:

1. **\(-12.5\)**: \( -12.5 \) is not less than or equal to \(-13.5\) (false).
2. **\(-13\)**: \( -13 \) is not less than or equal to \(-13.5\) (false).
3. **\(-12\)**: \( -12 \) is not less than or equal to \(-13.5\) (false).
4. **\(-13.5\)**: \( -13.5 \) is equal to \(-13.5\) (true).

Thus, the value that makes the inequality true is:

**\(-13.5\)**.
Answered by GPT-4o mini
To find which of the given equations has the solution set \(\{-3\}\), we will substitute \(x = -3\) into each equation and see which one holds true.

1. **Equation**: \(4x + 8 = 4\)
\[
4(-3) + 8 = 4 \\
-12 + 8 = 4 \\
-4 \neq 4 \quad \text{(not true)}
\]

2. **Equation**: \(4x - 8 = 4\)
\[
4(-3) - 8 = 4 \\
-12 - 8 = 4 \\
-20 \neq 4 \quad \text{(not true)}
\]

3. **Equation**: \(4x + 8 = -4\)
\[
4(-3) + 8 = -4 \\
-12 + 8 = -4 \\
-4 = -4 \quad \text{(true)}
\]

4. **Equation**: \(4x - 8 = -4\)
\[
4(-3) - 8 = -4 \\
-12 - 8 = -4 \\
-20 \neq -4 \quad \text{(not true)}
\]

The only equation that is true when \(x = -3\) is:

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

So, the answer is the third option:

**4x + 8 = -4**.
Answered by GPT-4o mini
To solve the compound inequality \(-2 \leq 2x + 13 - 4 < 1\), we can break it into two parts and solve each part separately.

1. **Simplify the compound inequality**:
\[
-2 \leq 2x + 9 < 1
\]

This means we have two inequalities to solve:
- \(-2 \leq 2x + 9\)
- \(2x + 9 < 1\)

### Solving the first inequality:

\[
-2 \leq 2x + 9
\]

Subtract \(9\) from both sides:
\[
-2 - 9 \leq 2x \\
-11 \leq 2x
\]

Now divide by \(2\):
\[
-\frac{11}{2} \leq x \quad \text{or} \quad x \geq -\frac{11}{2}
\]

### Solving the second inequality:

\[
2x + 9 < 1
\]

Subtract \(9\) from both sides:
\[
2x < 1 - 9 \\
2x < -8
\]

Now divide by \(2\):
\[
x < -4
\]

### Combine the results:

We have:
\[
-\frac{11}{2} \leq x < -4
\]

The value \(-\frac{11}{2}\) is \(-5.5\), so we rewrite the result as:
\[
-5.5 \leq x < -4
\]

### Final solution set:

Expressing this in set notation, we write:
\[
\{x : -5.5 \leq x < -4\}
\]

Since none of the responses appear to match this exact form, we can convert \(-5.5\) to \(-\frac{11}{2}\):

Thus, the final solution is:
\[
\{x: -\frac{11}{2} \leq x < -4\}
\]

From the provided responses, **none seem to match exactly**. However, if we compare values, we approximate which might resemble the closest form.

The closest option could possibly be considered **\(\{x: -\frac{3}{2} \leq x < 1\}\)**, but this doesn't match at all with derived results.

In this case, ensure to double-check options provided or hint back toward problem-specified interpretations.
Answered by GPT-4o mini
To identify the property of equality displayed in Step 3 of solving the equation \(9m + 4 - 7m = 5m + 73\), let's look closely at Step 3 itself:

In Step 3, you have:
\[
6m + 12 = 5m + 7
\]
And then you perform the operation:
\[
6m + 12 - 12 = 5m + 7 - 12
\]

This step involves subtracting \(12\) from both sides of the equation in order to isolate the terms involving \(m\).

### The property used here is:

**Subtraction Property of Equality**

This property states that if you subtract the same value from both sides of an equation, the two sides remain equal.

Thus, the correct response identifying the property of equality displayed in Step 3 is:

**Subtraction Property of Equality**.