Question

Which of the following is an equivalent form of the compound inequality −22 > −5x − 7 ≥ −3?

Answer 1

The answer is: [A]: " −5x − 7 < −22 and −5x − 7 ≥ −3 "
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Explanation:
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We are the given the following compound inequality:
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" −22 > −5x − 7 ≥ −3 "
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Consider Answer choice: [A]: −5x − 7 < −22 and −5x − 7 ≥ −3
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Do both of these given in "Answer choice [A]" hold true?
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  First, does: −5x − 7 < −22 hold true?  
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Yes, because, in the compound inequality given:
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−22 > −5x − 7 ; which is another way of saying that:
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 −5x − 7 < −22 .
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Second does:  "−5x − 7 ≥ −3 " , hold true?
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Yes, because, in the compound equality given, 
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−5x − 7 ≥ −3 ; which explicitly tells us that it holds true.
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Thus, "Answer choice: [A]: −5x − 7 < −22 and −5x − 7 ≥ −3 " is a correct answer.
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There could be other correct answers, so let us examine the other answer choices:
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Consider "Answer choice: [B]: −5x − 7 > −22 and −5x − 7 ≥ −3 ;
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Do EACH of these hold true in "Answer choice "B"?
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First let's examine:
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 −5x − 7 > −22 ; Does this hold true?
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No! Because in the compound inequality given,
−5x − 7 < −22 ; so we already know that "Answer choice: [B]" is incorrect; and thus, it makes no difference whether the other part of "answer choice [B]" is correct or incorrect.
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Let us consider: "Answer choice: [C]: 5x > −22 and −7 ≥ −3" .
 Do these hold true with respect to our given compound inequality?
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No; since NEITHER one of of this 2 (two) components given are are part of our compound inequality given.
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Furthermore, the particular component given: " −7 ≥ −3" does not make sense. "Negative 7" is not 'greater than or equal to' "Negative 3".   -7 ≠ -3 . Additionally, "-7" is LESS THAN—not GREATER THAN—"-3". 
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Also, "−5x > −22" is NOT part of the compound inequality given. So, "Answer Choice: [C]:  −5x > −22 and −7 ≥ −3" ; is INCORRECT.
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Additional note: " −5x < −22" is NOT part of the compound inequality given, either.
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The [incorrect] "Answer Choice: [C]:  "−5x > −22 and −7 ≥ −3" ; can be somewhat deceptive.

The compound inequality given:
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" −22 > −5x − 7 ≥ −3 "  ;   should be viewed as:
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" −22 > (−5x − 7) ≥ −3 " ;
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and the term, "(−5x − 7)" should be treated as one separate term—just like the terms: "(-22)", and "(-3)".
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Let us consider "Answer choice: [D]: −5x − 7 < −22 and −5x − 7 ≤ −3 " .
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Start with: −5x − 7 < −22 .  Does this hold true for our original compound inequality? 
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Yes!  (Since in our original compound inequality; "−22 > −5x − 7" ;
 and this component of 'Answer choice: [D]' — "−5x − 7 < −22 "— is another way of saying that: "−22 > −5x − 7" (as denoted in our original compound inequality).
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Now, we shall examine the second component of
'Answer choice: [D]' — "−5x − 7 ≤ −3 ".
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Does this hold true for our original compound inequality? 
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No!  This second component given in our 'Answer choice'—is:
 "-5x − 7 ≤ −3" ; and in our original compound inequality: 
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"−5x − 7 ≥ −3 " ! 
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As such, since one of the compounds in 'Answer choice: [D]' is incorrect; 
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Answer choice: [D]: "−5x − 7 < −22 and −5x − 7 ≤ −3" ;  is incorrect.
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