The answer is: [A]: " −5x − 7 < −22 and −5x − 7 ≥ −3 "
Explanation:
We are the given the following compound inequality:
" −22 > −5x − 7 ≥ −3 "
Consider Answer choice: [A]: −5x − 7 < −22 and −5x − 7 ≥ −3
Do both of these given in "Answer choice [A]" hold true?
First, does: −5x − 7 < −22 hold true?
Yes, because, in the compound inequality given:
−22 > −5x − 7 ; which is another way of saying that:
−5x − 7 < −22 .
Second does: "−5x − 7 ≥ −3 " , hold true?
Yes, because, in the compound equality given,
−5x − 7 ≥ −3 ; which explicitly tells us that it holds true.
Thus, "Answer choice: [A]: −5x − 7 < −22 and −5x − 7 ≥ −3 " is a correct answer.
There could be other correct answers, so let us examine the other answer choices:
Consider "Answer choice: [B]: −5x − 7 > −22 and −5x − 7 ≥ −3 ;
Do EACH of these hold true in "Answer choice "B"?
First let's examine:
−5x − 7 > −22 ; Does this hold true?
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.
Let us consider: "Answer choice: [C]: 5x > −22 and −7 ≥ −3" .
Do these hold true with respect to our given compound inequality?
No; since NEITHER one of of this 2 (two) components given are are part of our compound inequality given.
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".
Also, "−5x > −22" is NOT part of the compound inequality given. So, "Answer Choice: [C]: −5x > −22 and −7 ≥ −3" ; is INCORRECT.
Additional note: " −5x < −22" is NOT part of the compound inequality given, either.
The [incorrect] "Answer Choice: [C]: "−5x > −22 and −7 ≥ −3" ; can be somewhat deceptive.
The compound inequality given:
" −22 > −5x − 7 ≥ −3 " ; should be viewed as:
" −22 > (−5x − 7) ≥ −3 " ;
and the term, "(−5x − 7)" should be treated as one separate term—just like the terms: "(-22)", and "(-3)".
Let us consider "Answer choice: [D]: −5x − 7 < −22 and −5x − 7 ≤ −3 " .
Start with: −5x − 7 < −22 . Does this hold true for our original compound inequality?
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).
Now, we shall examine the second component of
'Answer choice: [D]' — "−5x − 7 ≤ −3 ".
Does this hold true for our original compound inequality?
No! This second component given in our 'Answer choice'—is:
"-5x − 7 ≤ −3" ; and in our original compound inequality:
"−5x − 7 ≥ −3 " !
As such, since one of the compounds in 'Answer choice: [D]' is incorrect;
Answer choice: [D]: "−5x − 7 < −22 and −5x − 7 ≤ −3" ; is incorrect.
