Question
Explain why the equation 6|x| + 25 = 15 has no solution.
When one solves, they arrive at a step where |x| is equal to a fraction that may not be represented as an integer. Since | x| must be an integer, there is no solution.
The statement is false. There is a solution.
When one solves, they arrive at a step where x is equal to a negative number. Since x can never be negative inside of the absolute value bars, there is no solution.
When one solves, they arrive at a step where |x| is equal to a negative number. Since | x| can never be negative, there is no solution.
When one solves, they arrive at a step where |x| is equal to a fraction that may not be represented as an integer. Since | x| must be an integer, there is no solution.
The statement is false. There is a solution.
When one solves, they arrive at a step where x is equal to a negative number. Since x can never be negative inside of the absolute value bars, there is no solution.
When one solves, they arrive at a step where |x| is equal to a negative number. Since | x| can never be negative, there is no solution.
Answers
GPT-4o mini
To determine why the equation \( 6|x| + 25 = 15 \) has no solution, let's solve it step-by-step:
1. **Isolate the absolute value term:**
\[
6|x| + 25 = 15
\]
Subtract 25 from both sides:
\[
6|x| = 15 - 25
\]
\[
6|x| = -10
\]
2. **Divide by 6:**
\[
|x| = \frac{-10}{6}
\]
Simplifying the fraction gives:
\[
|x| = -\frac{5}{3}
\]
At this stage, we find that \( |x| \) is equal to \(-\frac{5}{3}\). However, we know that the absolute value of any real number \( x \) (denoted as \( |x| \)) is always non-negative; it cannot be negative.
Since we've arrived at an equation stating that \( |x| \) equals a negative number \(-\frac{5}{3}\), we conclude that this scenario is impossible.
Thus, the correct reasoning for the conclusion that there is no solution is captured by the statement:
"When one solves, they arrive at a step where \( |x| \) is equal to a negative number. Since \( |x| \) can never be negative, there is no solution."
This accurately reflects the rules of absolute values and confirms that the equation \( 6|x| + 25 = 15 \) has no solution.
1. **Isolate the absolute value term:**
\[
6|x| + 25 = 15
\]
Subtract 25 from both sides:
\[
6|x| = 15 - 25
\]
\[
6|x| = -10
\]
2. **Divide by 6:**
\[
|x| = \frac{-10}{6}
\]
Simplifying the fraction gives:
\[
|x| = -\frac{5}{3}
\]
At this stage, we find that \( |x| \) is equal to \(-\frac{5}{3}\). However, we know that the absolute value of any real number \( x \) (denoted as \( |x| \)) is always non-negative; it cannot be negative.
Since we've arrived at an equation stating that \( |x| \) equals a negative number \(-\frac{5}{3}\), we conclude that this scenario is impossible.
Thus, the correct reasoning for the conclusion that there is no solution is captured by the statement:
"When one solves, they arrive at a step where \( |x| \) is equal to a negative number. Since \( |x| \) can never be negative, there is no solution."
This accurately reflects the rules of absolute values and confirms that the equation \( 6|x| + 25 = 15 \) has no solution.