Which is the solution of -(6)/(x) -(x-2)/(4) \ \textgreater \ (3-x)/(3)? Multiple choice question. A) x= −6, x= 12, or x ≠ 0 B) −6 < x < 0 or x > 12 C) x < −6 or x &…

Question

Which is the solution of
`-(6)/(x) -(x-2)/(4) \ \textgreater \ (3-x)/(3)`?

Multiple choice question.


A)

x= −6, x= 12, or x ≠ 0


B)

−6 < x < 0 or x > 12


C)

x < −6 or x > 12


D)

−12 < x < 0 or x > 6

Answer 1

• C) x < −6 or x > 12

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Given inequality:

• - 6/x - (x - 2)/4 > (3 - x)/3

Consider x ≠ 0 and multiply all terms by x, 4 and 3:

• - 6*12 - 3x(x - 2) > 4x(3 - x)
• -72 - 3x² + 6x > 12x - 4x²
• 4x² - 3x² + 6x - 12x - 72 > 0
• x² - 6x - 72 > 0
• x² - 12x + 6x - 72 > 0
• x(x - 12) + 6(x - 12) > 0
• (x + 6)(x - 12) > 0

The x-intercepts are:

• x = - 6 and x = 12

This quadratic function has a positive leading coefficient and two zeros, and hence is positive when:

• x < - 6 and x > 12, the x = 0 is excluded from the given interval, therefore the above is the solution.

The matching choice is C.