Solve each equation for x, where x is restricted to the given interval.

y = √2 + 3 sec 2x, for x in [0, π/4) ⋃ (π/4, π/2]

Video transcript

Hey, everyone here we are asked to solve the following equation for X. If X lies in the interval from open bracket zero to pi divided by eight closed parentheses. And from open parentheses, pi divided by 82 pi divided by four closed bracket. Here we are given the equation Y is equal to the square root of five plus seven multiplied by second of four multiplied by X. Here we have four answer choice options, answer choices A through D, each of which contain an equation where X is solved for and includes the second inverse function of some fraction. So to begin solving our given equation for X, we can start by simply rearranging so that our X term is on the left hand side. And so doing this, our equation is now the square root of five plus seven multiplied by second of four, multiplied by X is equal to Y. And so now we just want to begin solving for X. And so we can start by moving over this square root of five over to the right hand side. And so that means we need to subtract the square root of five from both sides of the equation. And now rewriting, we have seven multiplied by second of four, multiplied by X is equal to Y minus the square root of five. And now continuing to solve four, X, we need to get rid of this coefficient of the second function. And so that means we need to divide both sides of the equation by the coefficient which is seven. So we divide both sides by seven and this gives us second of four, multiplied by X is equal to Y minus the square root of five all divided by seven. And so now continuing solving for X, we can notice that our next step here, we notice the X term is within the second function. So we need to get rid of the second function. And we do this by taking the second inverse of both sides of the equation. And so doing this, our new equation is four multiplied by X is equal to seek an inverse of Y minus the square root of five all divided by seven. And with this, we are left with four multiplied by X on the left hand side. So now we just need to get rid of this coefficient. And so just like earlier, we divide both sides by the coefficient which in this case is four. So dividing both sides by four, we get X is equal to one divided by four multiplied by see inverse of Y minus the square root of five. All divided by seven. And now we can see we have finished solving for X. So this is our final answer which leaves us with answer choice D where again we have X is equal to one divided by four, multiplied by second inverse of Y minus the square root of five, all divided by seven. Thanks for watching. I hope you found this video helpful and I'll see you again next time.