Question

Which inequalities describe the given graph?

Select each correct answer.

​3(4−x)<30​

2/3x+9>5

−2(9−x)<−6

2x−5>7

Answer 1

The correct answers to the given graph/number line:  " x > -6 " ;

are: Answer choices:  [A]: ​" 3(4 − x) < 30​ " ;
                             and:   [B]: " $\frac{2}{3}$ x + 9 > 5 " .
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Step-by-step 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]:  " $\frac{2}{3}$ x + 9 > 5 " ;  To simplify:
Subtract "9" from each side:
  " $\frac{2}{3}$ x + 9 − 9 > 5 − 9 " ; to get:  " $\frac{2}{3}$ x > -4 ";  
Now, multiply the entire inequality by "3";
⇒  3*[$\frac{2}{3}$ 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]: " $\frac{2}{3}$ x + 9 > 5 " .

Hope this is helpful.  Best wishes to you in your academic endeavors!
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