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

Question

Responses

{x=3}
left brace x equals 3 right brace

(3)
left parenthesis 3 right parenthesis

{3}
left brace 3 right brace

​ (x=3)

Answer 1

To express the solution set for the equation \( 31 - 9x = 4 \), we first need to solve for \( x \).

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

2. Divide by -9: \[ x = 3 \]

Now, to express the solution set:

• The solution \( x = 3 \) is a specific number, so we can represent the solution set as \( { 3 } \).

Therefore, the correct set notation that expresses the solution set is: \[ { 3 } \]

So the answer is {3}.

Answer 2

Trey, 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?

(1 point)
Responses

Trey
Trey

None of them are correct.
None of them are correct.

Cho
Cho

Amelia
Amelia
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Answer 3

To determine the correct solution set for the equation \( x^2 + 9 = 45 \), we need to first solve for \( x \).

1. Start by isolating \( x^2 \): \[ x^2 + 9 = 45 \implies x^2 = 45 - 9 \] \[ x^2 = 36 \]

2. Next, take the square root of both sides: \[ x = \pm 6 \]

This means the two solutions are \( x = 6 \) and \( x = -6 \).

Now, let's evaluate the answers given by Trey, Amelia, and Cho:

• Trey: {6} (incorrect, as it doesn't include \(-6\))
• Amelia: {-6, 6} (correct, as it includes both solutions)
• Cho: {-6} (incorrect, as it doesn't include 6)

Thus, the person who has found the correct solution set is Amelia.

Answer 4

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

​ {x=−1,x=1}
​ left brace x equals negative 1 comma x equals 1 right brace

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

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

{−1,1}

Answer 5

To express the solution set for the values that make the equation \( 3x^2 + 7 = 10 \) true, we note that the solutions are \( x = -1 \) and \( x = 1 \).

The correct set notation to represent these solutions is:

\[ { -1, 1 } \]

This means the correct response is {−1, 1}.

Answer 6

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 greater than 15 right brace

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

{b:b>3}
left brace b colon b greater than 3 right brace

{b:b<3

Answer 7

To express the solution set of the inequality \( 3(b - 4) + 2b > 3 \), we first need to check if the statement that the solutions are the set of values of \( b \) greater than 3 is correct.

Let's solve the inequality:

1. Distribute the 3 into \( (b - 4) \): \[ 3b - 12 + 2b > 3 \]

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

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

4. Divide by 5: \[ b > 3 \]

So, the solutions to the inequality are indeed \( b > 3 \).

The correct set notation to express these solutions is: \[ { b : b > 3 } \]

Thus, the appropriate response is {b:b>3}.

Answer 8

What value falls in the solution set of the inequality −3(4k+1)<15 ?(1 point)
Responses

−1
negative 1

−4
negative 4

−32
Start Fraction negative 3 over 2 End Fraction

−5

Answer 9

To find a value that falls in the solution set of the inequality \(-3(4k + 1) < 15\), we need to solve the inequality for \(k\).

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

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

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

Now we need to evaluate the options provided to see which of them is greater than \(-\frac{3}{2}\) (which is \(-1.5\)):

• −1: This is greater than \(-\frac{3}{2}\).
• −4: This is less than \(-\frac{3}{2}\).
• −32: This is less than \(-\frac{3}{2}\).
• −5: This is less than \(-\frac{3}{2}\).

The only value that falls in the solution set \( k > -\frac{3}{2} \) is −1.

So, the correct answer is −1.

Answer 10

What value falls in the solution set of the inequality −3(4k+1)<15 ?(1 point)
Responses

−1
negative 1

−4
negative 4

−3/2
Start Fraction negative 3 over 2 End Fraction

−5

Answer 11

Let's solve the inequality \(-3(4k + 1) < 15\) step by step and determine which of the given values fall in the solution set.

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

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

3. Divide by -12 (remember to reverse the inequality sign): \[ k > -\frac{3}{2} \]

Now we know that the solution set is \( k > -\frac{3}{2} \) (which is equivalent to \( k > -1.5 \)).

Next, let's evaluate the options:

• −1: This is greater than \(-\frac{3}{2}\) (valid solution).
• −4: This is less than \(-\frac{3}{2}\) (not a valid solution).
• −3/2: This is exactly \(-1.5\), but not greater than \(-\frac{3}{2}\) (not a valid solution).
• −5: This is less than \(-\frac{3}{2}\) (not a valid solution).

The only value that falls in the solution set of the inequality \( k > -\frac{3}{2} \) is −1.

So the correct answer is −1.