Question

x-1/x+4 = 1/4, find x​

Answer 1

Answer:

If x + 1/x = 4, then x =?

Let’s expand this a bit. It is clear that for any nonzero real number, x, x and 1x have the same sign. Therefore, f(x)=x+1x is negative when x < 0 and positive when x > 0. Furthermore, f(-x) = -f(x), so that f is an odd function. We need only concern ourselves with positive values of x, and the behavior for negative values will follow logically.

Now, we see that when x > 0, either x or its reciprocal must be greater than 1. In fact, the minimum value for f(x) when x > 0 is 2.

Let a≥2 .

x+1x=a

x2+1=ax

x2−ax+1=0

By the Quadratic Formula,

x=a±a2−4−−−−−√2

Observe that when a = 2, there is exactly one solution: x = 1.

When a > 2, there two distinct real solutions.

In particular, when a = 4, we have

x=4±42−4−−−−−√2

x=4±16−4−−−−−√2

x=4±12−−√2

x=4±23–√2

x=2(2±3–√)2

x=2±3–√

Answer 2

Question

• x-1/x+4 = 1/4, find x

Solution

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⟼$\tt\bold{ \frac{x - 1}{x + 4} } = \frac{1}{4}$

⟼$\tt\bold{cross \: multiply \: }$

⟼$\tt\bold{4(x - 1) = 1(x + 4)}$

⟼$\tt\bold{4x - 4 = x + 4}$

⟼$\tt\bold{4x - x = 4 + 4}$

⟼$\tt\bold{3x = 8}$

⟼$\tt\bold{x = \frac{8}{3} }$

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