The correct answers to the given graph/number line: " x > -6 " ;
are: Answer choices: [A]: " 3(4 − x) < 30 " ;
and: [B]: "
x + 9 > 5 " .
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Explanation:
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Note: The "given graph" is a "number line" that represents:
" x > - 6 " .
The "negative 6"; that is: "-6" is "circled"; which means "does not equal: ("-6"); there is a line, or geometrically speaking, a "ray" ; that continues to the side going forward to all value greater the "-6" .
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The question asks:
What inequalities describe the given {graph—in our case, number line.}.
So, there are 4 (four) answer choices.
We are to determine which Answer choice(s) are equal to:
" x > - 6 " .
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Choice [A]: " 3(4 − x) < 30 " ;
To simplify:
First, let us divide each side of this 'inequality; by "3":
" [3(4 − x)] / 3 < [30/3] " to get: " 4 − x < 10 " ;
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We can multiply each side of this inequality by "-1" ;
" -1(4 − x) < 10(-1) " ;
Note: - 1(x − y) = y − x ;
and when each side of an Inequality, specifically, is multiplied or divided by a 'negative number'; we change the 'direction' of the inequality ;
So we simplify our inequality:
" x − 4 > -10 " ; Add "4" to both sides of this inequality:
" x − 4 + 4 > -10 + 4 " ; to get: " x > -6 " ; which is a correct answer.
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Choice: [B]: "
x + 9 > 5 " ; To simplify:
Subtract "9" from each side:
"
x + 9 − 9 > 5 − 9 " ; to get: "
x > -4 ";
Now, multiply the entire inequality by "3";
⇒ 3*[
x] > 3*(-4) ";
to get: " 2x > -12 "; Now, divide Each side by "2" :
" 2x /2 > -12 / 2 "; to get: " x > -6 " ; which is a correct answer!
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Choice [C]: " -2(9 − x) < -6 " ; To simplify: Divide each side by "-2" ;
→ " [-2(9 − x)] /-2] < -6/-2 " ;
Note: When multiplying and dividing both sides of an 'inequality' by a 'negative number; the "sign/direction" of the inequality changes.
→ to get: " 9 − x > 3 "; Now, multiply each side by "(-1)";
→ " -1(9 − x) > (3*-1); Note, we are multiplying by a negative number, so we shall reverse the direction/sign of the inequality.
Also: Note: " -(x − y) = y − x) " ;
→ So, we can simplify:
" x − 9 < -3 " ;
Now, add "9" to each side:
" x − 9 + 9 < -3 + 9 "; to get: " x < -6 " ; which is not a correct answer.
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Choice: [D]: " 2x − 5 > 7 "; Add "5" to each side of the inequality:
" 2x − 5 + 5 > 7 + 5 " ; to get:
2x > 12 ; Now divide each side of the inequality by "2" ;
to isolate "x" on one side of the equation; & to solve for "x" ;
2x / 2 > 12/2 ;
to get: "x > 6 " ; which is not a correct answer.
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The correct answers to the given graph/number line: " x > -6 " ;
are: Answer choices: [A]: " 3(4 − x) < 30 " ;
and: [B]: "
x + 9 > 5 " .
Hope this is helpful. Best wishes to you in your academic endeavors!
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