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Which set notation correctly expresses the solution set to the equation 31−9x=4 ?(1 point)

Responses:
{x=3}
{3}
​ (x=3)
(3)

9 answers

To solve the equation \( 31 - 9x = 4 \):

  1. Subtract 31 from both sides: \[ -9x = 4 - 31 \] \[ -9x = -27 \]

  2. Divide both sides by -9: \[ x = \frac{-27}{-9} \] \[ x = 3 \]

The solution set to the equation is just the value of \( x \), which is 3.

Among the provided choices, the correct notation that expresses the solution set is: {3}

This indicates that the solution set contains the single element 3.

rey, Amelia, and Cho are discussing the solution set to the equation x2+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?
Responses:
None of them are correct.
Trey
Cho
Amelia

To find the solution set for the equation \( x^2 + 9 = 45 \), we first solve for \( x \):

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

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

So, the solutions are \( x = 6 \) and \( x = -6 \). This means the correct solution set is \( {-6, 6} \).

Amelia says the solution is {−6, 6}, which is correct.

Thus, the answer is: Amelia.

The values −1 and 1 make the equation 3x2+7=10 true. Use set notation to express the solution to the equation.(1 point)
Responses:
(−1,1)
​ {x=−1,x=1}
{−1,1}
(x=−1,x=1)

The correct solution set to the equation, which contains the values \(-1\) and \(1\), can be expressed in set notation.

Among the options provided, the correct notation for the solution set is: {−1, 1}

This notation indicates that the solution set contains the elements \(-1\) and \(1\).

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}
{b:b<3}
{b:b>3}
{b:b<15}

To solve the inequality \( 3(b - 4) + 2b > 3 \), we can simplify it 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 by 5: \[ b > 3 \]

The solution set is the set of values of \( b \) that are greater than 3.

The correct set notation that expresses these solutions 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
Responses:
−1
−3/2
−5
−4

To solve the inequality \( -3(4k + 1) < 15 \), let's solve it step by step:

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

  2. Add 3 to both sides: \[ -12k < 15 + 3 \] \[ -12k < 18 \]

  3. Divide both sides by -12 (remember to flip the inequality when dividing by a negative number): \[ k > \frac{18}{-12} \] \[ k > -\frac{3}{2} \]

Thus, the solution to the inequality is \( k > -\frac{3}{2} \).

Now, we need to find which of the provided values falls in the solution set \( k > -\frac{3}{2} \):

  • −1: This is greater than \(-\frac{3}{2}\) (approximately \(-1.5\)), so it satisfies the inequality.
  • −3/2: This is equal to \(-\frac{3}{2}\), thus does not satisfy \( k > -\frac{3}{2} \).
  • −5: This is less than \(-\frac{3}{2}\), so it does not satisfy the inequality.
  • −4: This is also less than \(-\frac{3}{2}\) and does not satisfy the inequality.

Therefore, the value that falls in the solution set of the inequality \( -3(4k + 1) < 15 \) is: −1.