Question


Trey, Amelia, and Cho are discussing the solution set to the equation x^2+9=45.

Trey says the solution is {6}.

Amelia says the solution is {−6,6}.

Cho says the solution is {−6}.

Which person has found the correct solution set?

(1 point)
Responses

Amelia
Amelia

Cho
Cho

Trey
Trey

None of them are correct.
Answer
The values −1 and 1 make the equation 3x^2+7=10 true. Use set notation to express the solution to the equation.(1 point)
Responses

(−1,1)
left parenthesis negative 1 comma 1 right parenthesis

{−1,1}
left brace negative 1 comma 1 right brace

(x=−1,x=1)
left parenthesis x equals negative 1 comma x equals 1 right parenthesis

​ {x=−1,x=1}
Answer
The solution set of the inequality 3(b−4)+2b>3 is the set of values of b greater than three. Which set notation correctly expresses the solutions?(1 point)
Responses

{b:b<15}
left brace b colon b less than 15 right brace

{b:b>15}
left brace b colon b greater than 15 right brace

{b:b<3}
left brace b colon b less than 3 right brace

{b:b>3}
Answer
What value falls in the solution set of the inequality −3(4k+1)<15?(1 point)
Responses

−1
negative 1

−5
negative 5

−4
negative 4

−3/2
Answered by GPT-4o mini
To solve the equation \(x^2 + 9 = 45\), we need to isolate \(x\). We can do this by following these steps:

1. Subtract 9 from both sides:
\[
x^2 = 45 - 9
\]
\[
x^2 = 36
\]

2. Take the square root of both sides:
\[
x = \pm \sqrt{36}
\]
\[
x = \pm 6
\]

This gives us two solutions: \(x = 6\) and \(x = -6\).

Now let's evaluate the responses:

- Trey says the solution is \{6\}. (Incorrect, as we also have \(-6\))
- Amelia says the solution is \{-6, 6\}. (Correct, as both values are solutions)
- Cho says the solution is \{-6\}. (Incorrect, as we also have \(6\))

Therefore, the correct answer is **Amelia**.
Answered by GPT-4o mini
To express the solution set for the equation \(3x^2 + 7 = 10\), we found that the values \(-1\) and \(1\) satisfy the equation. In set notation, a solution set containing distinct elements is typically represented with braces.

Thus, the correct representation of the solution set is \(\{-1, 1\}\).

So the correct response is **\{−1, 1\}**.
Answered by GPT-4o mini
To determine the solution set of the inequality \(3(b - 4) + 2b > 3\), let’s first solve the inequality step by step.

1. Distribute the \(3\):
\[
3b - 12 + 2b > 3
\]

2. Combine like terms:
\[
5b - 12 > 3
\]

3. Add \(12\) to both sides:
\[
5b > 15
\]

4. Divide both sides by \(5\):
\[
b > 3
\]

The solution set is all values of \(b\) greater than \(3\).

The correct representation of this in set notation is:
\[
\{b : b > 3\}
\]

Thus, the correct response is **\{b:b>3\}**.
Answered by GPT-4o mini
To solve the inequality \(-3(4k + 1) < 15\), we will proceed with the following steps:

1. Distribute the \(-3\):
\[
-12k - 3 < 15
\]

2. Add \(3\) to both sides:
\[
-12k < 18
\]

3. Divide both sides by \(-12\). Remember that dividing by a negative number reverses the inequality:
\[
k > -\frac{3}{2}
\]

So the solution set consists of all values of \(k\) that are greater than \(-\frac{3}{2}\).

Now, we will check which of the given options fall within this solution set:

- \(-1\): This value is greater than \(-\frac{3}{2}\), so it is in the solution set.
- \(-5\): This value is less than \(-\frac{3}{2}\), so it is not in the solution set.
- \(-4\): This value is less than \(-\frac{3}{2}\), so it is not in the solution set.
- \(-\frac{3}{2}\): This value is equal to \(-\frac{3}{2}\) and does not satisfy \(k > -\frac{3}{2}\), so it is not in the solution set.

Thus, the value that falls within the solution set is **\(-1\)**.