SOLUTION: 1) Solve 2(√4x+5) - (√3x+1) = 6, where x is an integer. 2) Given that sinx = -3/5, where 180° ≤ x ≤ 270° and cos y = - 15/17 , where 90° ≤ y ≤ 180°. find the val

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Question 1210429: 1) Solve 2(√4x+5) - (√3x+1) = 6, where x is an integer.
2) Given that sinx = -3/5, where 180° ≤ x ≤ 270° and cos y = - 15/17 , where 90° ≤ y ≤ 180°. find the value of tan(x-y).
Found 3 solutions by ikleyn, greenestamps, n2:
Answer by ikleyn(53354)   (Show Source): You can put this solution on YOUR website!
.
1) Solve 2(√4x+5) - (√3x+1) = 6, where x is an integer.
2) Given that sinx = -3/5, where 180° ≤ x ≤ 270° and cos y = - 15/17 , where 90° ≤ y ≤ 180°.
find the value of tan(x-y).
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


        I will solve here problem  (2),  ONLY. 


Since sin(x) = -3/5 in QIII, it implies  cos(x) =  = -4/5.


Since cos(y) = -15/17 in QII,  it implies  sin(y) =  = 8/17.


Therefore  tan(x) =  =  = 3/4,

           tan(y) =  =  = -8/15.


Now apply the formula for tan(x-y)

    tan(x-y) = .


In the numerator, we have  

    tan(x) - tan(y) =  =  =  =  = .


In the denominator, we have

    1 + tan(x)*tan(y) =  =  = .


Thus finally  tan(x-y) =  = .


ANSWER.  tan(x-y) = .

Part (2) is completed.


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Answer by greenestamps(13250)   (Show Source): You can put this solution on YOUR website!


For the first problem....

NOTE: Don't use "√" when posting problems involving square roots -- or, if you do, enclose the expressions under the radicals in parentheses. "2(√4x+5)" looks like , when apparently the intended expression is 2(√(4x+5)) =

Outline for a formal solution....
(1) rewrite the equation with one radical on each side
(2) square both sides of that equation; you will still have a radical on one side
(3) isolate that radical on one side and again square both sides of the equation, resulting finally in an equation without any radicals

If you do all that algebra correctly, you will end up with this quadratic equation:

Options for what to do from here....

(1) solve that equation by factoring (have fun!!!)
(2) solve the equation using the quadratic formula (perhaps even MORE fun!!!!)
(3) solve the equation using a graphing calculator (e.g., TI84 or equivalent; or online at desmos.com)
Option (3) is much less work than either of the other options....

But if you are going to solve the problem using technology, why bother with the algebra outlined above? Simply use a graphing utility to graph the expressions on each side of the original equation and find the x value where the graphs intersect.

Finally, what is the easiest (and by far the fastest) way to solve the problem?

Assume the answer is a whole number; ignore ALL of the above and look for whole number values of x for which (4x+5) and (3x+1) are both perfect squares.

ANSWER: x = 5

CHECK: 2*sqrt(4x+5)-sqrt(3x+1) = 2*sqrt(25)-sqrt(16) = 2(5)-4 = 10-4 = 6
Answer by n2(19)   (Show Source): You can put this solution on YOUR website!
.
1) Solve 2(√4x+5) - (√3x+1) = 6, where x is an integer.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


Tutor @greenestamps made a great step forward, guessing or finding mentally
one integer solution  x = 5.

But when a solution to a Math problem,  or a math equation is found by the  " trial and error "
guessing method,  always a question arises - whether this solution is unique or whether other solutions exist.

There is one brilliant reasoning,  which may help in some cases,
and it will help to answer for this given equation.

Notice that in the left side of the equation the term    grows  FASTER  than
the other term,  .     (This is obvious).

AFORTIORI,  the term    grows even  FASTER  than the other term,  .

It means that the left side of the given equation is a monotonic function of  x
(monotonically increases for positive values of  x).

Hence,  this given equation,  having a constant value in its right side,  has a  UNIQUE  solution.

In other words - if it has one solution,  then this solution is unique.

Reasoning this way,  we transform our  " trial and error "  guessing method into  MATHEMATICALLY  STRICT   .

So,  x = 5  is the unique real and integer solution to this given equation. 

                             Z Z Z  !


       Be careful !  That is true that in the positive 'x'-domain the left side monotonically increases.

       But what about negative 'x' ? - In this problem, the domain of negative 'x' is small and does not 
       make influence to the final conclusion !



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