Question: Use the definition of absolute value and systems of inequalities to prove that for any real numbers x and c, and any positive real number δ, the given statement is true: | x − c | &lt; δ ⇔ x ∈ ( c − δ , c + δ )

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Question: Use the definition of absolute value and systems of inequalities to prove that for any real numbers x and c, and any positive real number δ, the given statement is true:  |  x  −  c  |  &amp;lt;  δ  ⇔  x  ∈  (  c  −  δ  ,  c  +  δ  )

Punktatsp · Accepted Answer

Like any bi-implication, it requires you to demonstrate two points:1. That if&amp;nbsp;|&amp;nbsp;x&amp;nbsp;−&amp;nbsp;c&amp;nbsp;|&amp;nbsp;&amp;lt;&amp;nbsp;δ, then&amp;nbsp;x&amp;nbsp;must be in the interval&amp;nbsp;(&amp;nbsp;c&amp;nbsp;−&amp;nbsp;δ&amp;nbsp;,&amp;nbsp;c&amp;nbsp;+&amp;nbsp;δ&amp;nbsp;); and2. That if&amp;nbsp;x&amp;nbsp;is in the interval&amp;nbsp;(&amp;nbsp;c&amp;nbsp;−&amp;nbsp;δ&amp;nbsp;,&amp;nbsp;c&amp;nbsp;+&amp;nbsp;δ&amp;nbsp;), then it must be the case that&amp;nbsp;|&amp;nbsp;x&amp;nbsp;−&amp;nbsp;c&amp;nbsp;|&amp;nbsp;&amp;lt;&amp;nbsp;δ.You are also told to use &quot;the definition of absolute value and systems of inequalities&quot; to support these claims.There are numerous ways to establish the fundamental qualities (typically, a small number of assertions may be used to establish all the others), so knowing which one is being utilized in your course will be crucial to how you show this. We won&#039;t try to predict what that means for your course.

Question: Use the definition of absolute value and systems of inequalities to prove that for any rea

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