text: Thread: Rewrite as a double inequality?
  
  1. Rewrite as a double inequality?
  
  I stumbled upon this question: "Rewrite |5x+7| < 10 as double inequality".
  
  My attempt:
  
  5x+7 < 10 and (-5x+7)<10
  5x < 3 and -5x<17
  x <3/5 and x>17/ 5
  
  Is that the solution? The given options are :
  
  a. -10> 5x +7 >10
  b. -10< 5x+7<10
  c. -10 < 5x+7>10
  d. 0<5x+7<10
  
  (I know c and a are wrong because of the direction of the greater than sign. To me b. seems right)
  
  2. Re: Rewrite as a double inequality?
  
  Originally Posted by sakonpure6
  I stumbled upon this question: "Rewrite |5x+7| < 10 as double inequality".
  This depends upon this theorem: $|a|\le |b|\text{ if and only if }-|b|\le a\le |b|$.
  
  So in this case $-10<5x+7<10$.
  
  3. Re: Rewrite as a double inequality?
  
  So the value of b needs to always be positive? Does this work |5x+7| < -10 ? No? May i please get the name of this theorem. Thank you!
  
  4. Re: Rewrite as a double inequality?
  
  Originally Posted by sakonpure6
  So the value of b needs to always be positive? Does this work |5x+7| < -10 ? No? May i please get the name of this theorem.
  It is such a basic property of the metric that it has no name.
explanation: The document is a thread from a forum where a user asks how to rewrite an absolute value inequality as a double inequality. The user provides their attempt at solving the problem and discusses the given options. Another user responds with a theorem that explains the correct way to rewrite the inequality and confirms the correct option. The thread also includes a follow-up question about the properties of the inequality.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: Thread: Rewrite as a double inequality?
  
  1. Rewrite as a double inequality?
  
  I stumbled upon this question: "Rewrite |5x+7| < 10 as double inequality".
  
  My attempt:
  
  5x+7 < 10 and (-5x+7)<10
  5x < 3 and -5x<17
  x <3/5 and x>17/ 5
  
  Is that the solution? The given options are :
  
  a. -10> 5x +7 >10
  b. -10< 5x+7<10
  c. -10 < 5x+7>10
  d. 0<5x+7<10
  
  (I know c and a are wrong because of the direction of the greater than sign. To me b. seems right)
  
  2. Re: Rewrite as a double inequality?
  
  Originally Posted by sakonpure6
  I stumbled upon this question: "Rewrite |5x+7| < 10 as double inequality".
  This depends upon this theorem: $|a|\le |b|\text{ if and only if }-|b|\le a\le |b|$.
  
  So in this case $-10<5x+7<10$.
  
  3. Re: Rewrite as a double inequality?
  
  So the value of b needs to always be positive? Does this work |5x+7| < -10 ? No? May i please get the name of this theorem. Thank you!
  
  4. Re: Rewrite as a double inequality?
  
  Originally Posted by sakonpure6
  So the value of b needs to always be positive? Does this work |5x+7| < -10 ? No? May i please get the name of this theorem.
  It is such a basic property of the metric that it has no name.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a thread from a forum where a user asks how to rewrite an absolute value inequality as a double inequality. The user provides their attempt at solving the problem and discusses the given options. Another user responds with a theorem that explains the correct way to rewrite the inequality and confirms the correct option. The thread also includes a follow-up question about the properties of the inequality.
  
  Label: question-and-answer