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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.

7 answers

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.

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}

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}.

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}

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}.

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

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 \(-1\).