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Use set notation to express the value(s) that make the equation 2x=20 true. Enter your response using set notation.(1 point)

11 answers

To solve the equation \(2x = 20\), we first divide both sides by 2:

\[ x = \frac{20}{2} = 10 \]

The solution to the equation is \(x = 10\).

In set notation, we express this solution as:

\[ {10} \]

Both −6 and 6 make the equation x^2/2=18 true. Use set notation to express these solutions. When entering your response, list elements in the set from least to greatest, with a comma between each element in the set.(1 point)

To solve the equation \(\frac{x^2}{2} = 18\), we first multiply both sides by 2:

\[ x^2 = 36 \]

Next, we take the square root of both sides:

\[ x = 6 \quad \text{or} \quad x = -6 \]

The solutions to the equation are \(6\) and \(-6\).

In set notation, listing the elements in order from least to greatest, we express this as:

\[ {-6, 6} \]

The solution to the inequality 4y+1>17 is y>4. Use set notation to express the values of y that make the inequality 4y+1>17 true.(1 point)

To express the solution to the inequality \(4y + 1 > 17\) in set notation, we first solve for \(y\):

  1. Subtract 1 from both sides: \[ 4y > 16 \]

  2. Divide both sides by 4: \[ y > 4 \]

The solution to the inequality is \(y > 4\).

In set notation, this is expressed as:

\[ (4, \infty) \]

This interval notation indicates that \(y\) can take any value greater than 4.

try again please

I apologize for any confusion in my previous response. In set notation, the values of \(y\) that satisfy the inequality \(4y + 1 > 17\), which simplifies to \(y > 4\), can be expressed as follows:

\[ { y \in \mathbb{R} \mid y > 4 } \]

This notation means "the set of all real numbers \(y\) such that \(y\) is greater than 4."

If you prefer interval notation (which is also a standard way to represent ranges), the solution is:

\[ (4, \infty) \]

However, since you requested set notation specifically, the first representation is the correct answer. Thank you for your understanding!

Sophie, Adola, and Gilly are debating potential values in the solution set of the compound inequality −2≤2x−2<8. Sophie says that −2 is a solution. Adola says that 5 is a solution. Gilly says that 2 is a solution. Which person is correct? Enter 1 for Sophie, 2 for Adola, or 3 for Gilly.(1 point)

To determine which person is correct, we first need to solve the compound inequality \(-2 \leq 2x - 2 < 8\).

We'll break it down into two parts:

  1. Solving the left part: \(-2 \leq 2x - 2\)

    Add 2 to both sides: \[ 0 \leq 2x \] Divide by 2: \[ 0 \leq x \quad \text{or} \quad x \geq 0 \]

  2. Solving the right part: \(2x - 2 < 8\)

    Add 2 to both sides: \[ 2x < 10 \] Divide by 2: \[ x < 5 \]

Combining both parts, we have: \[ 0 \leq x < 5 \]

Now we evaluate who among Sophie, Adola, and Gilly is correct:

  • Sophie says \(-2\) is a solution: This is incorrect because \(-2 < 0\).
  • Adola says \(5\) is a solution: This is incorrect because \(5\) is not less than \(5\) (not included).
  • Gilly says \(2\) is a solution: This is correct because \(0 \leq 2 < 5\).

Thus, Gilly is correct. The correct answer is:

3

Which option correctly describes the values that make the inequality 2m<14 true when written in set notation?

Option #1: {m:m<7}

Option #2: {7:7<m}

Option #3: {m<7:m}

Option #4: {m<7}

(1 point)
Option # correctly states the values that make the inequality true in set notation.

To solve the inequality \(2m < 14\), we first divide both sides by 2:

\[ m < 7 \]

Now, let's evaluate the provided options for correct set notation that describes the values of \(m\) making the inequality true:

  • Option #1: \({m: m < 7}\) - This option correctly indicates that \(m\) is any value less than 7. This is the correct set notation.

  • Option #2: \({7: 7 < m}\) - This option incorrectly suggests that the set contains \(7\) as an element, which is not correct since \(m\) should be less than \(7\).

  • Option #3: \({m < 7: m}\) - This option is incorrect and does not represent valid set notation.

  • Option #4: \({m < 7}\) - This option also does not represent values in correct set notation since it does not involve a relationship or a proper statement for \(m\).

Based on this analysis, Option #1: \({m: m < 7}\) is the correct choice.