please help me understand my teacher’s comments

/in /by

please help me understand my teacher’s comments

I thought I did the problem correct but the teacher’s feedback has me stumped can someone please explain to me what she is talking about and what I did wrong?

The problem I did:

9x^2+16−24x≥0

9x^2−24x+16≥0

9x^2−24x+16≥0

.(9x^2−12x−12x+16)≥0

((9x^2−12x)+(−12x+16))≥0

(3x(3x−4)−4(3x−4))≥0

(3x−4)(3x−4)≥0

(3x−4)1+1≥0

(3x−4)2≥0

(3x−4)2≥0

3x−4≥0

3x≥4

3x/3≥4/3

3 x/ 3 ≥4/3

x≥4/3

3x≥4

3x/3≥4/3

3 x/ 3 ≥4/3

x≥4/3

[4/3,Infinity)

 Teacher’s comments:

Watch out, because (3x-4)2 ≥ 0 does NOT imply that (3x – 4) ≥ 0. Also you are missing division symbols. 43 is not the same as 4/3.

When working with a polynomial inequality, first solve the related equation to get critical values. In this case, the equation (3x-4)2= 0 has the solution x = 4/3.

The value 4/3 splits up the x-axis into the intervals (- ∞ , 4/3) and (4/3, ∞).