Respuesta :

The solution to the problem is as follows:

4secx + 6 = −2

⟹secx = −2 = sec π/3

⟹x = 2nπ  ± π / 3, n∈W

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Answer:

[tex]x = \frac{2 \pi}{3} , \frac{4 \pi}{3}[/tex]

Step-by-step explanation:

Given the equation:

[tex]4 \sec x+6 = -2[/tex]

Subtract 6 from both sides we have;

[tex]4 \sec x =-8[/tex]

Divide both sides by 4 we have;

[tex]\sec x = -2[/tex]

We know that:

[tex]\sec x = \frac{1}{\cos x}[/tex]

then;

[tex]\cos x = -\frac{1}{2}[/tex]

⇒[tex]x = \cos^{-1} (-\frac{1}{2})[/tex]

Since, cos is negative in quadrant IInd and IIIrd quadrant  

⇒[tex]x = \pi -\frac{\pi}{3}, \pi +\frac{\pi}{3}[/tex]

⇒[tex]x = \frac{2 \pi}{3} , \frac{4 \pi}{3}[/tex]

therefore, the value of x over the interval [tex][0, 2 \pi][/tex] are:

[tex]x = \frac{2 \pi}{3} , \frac{4 \pi}{3}[/tex]