Solve X^5 +7X^3 +6X < 5X^4+ 7X^3 +2

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ishpiro | College Teacher | (Level 1) Educator

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`x^5 + 7x^3 + 6x < 5x^4 + 7x^3 + 2`

Let's start by putting all terms on the left-hand side and combining like terms:

`x^5+7x^3 + 6x - 5x^4 - 7x^3 - 2 <0`

`x^5 - 5x^4+6x - 2 <0`

To solve this inequality, we first need to solve the equation

`x^5 - 5x^4 + 6x - 2 = 0`

By inspection we can notice that x = 1 is one of the roots of this equation, so (x-1) will be one of the factors of the polynomial on the left side.

We can find another factor by synthetic division:

1 |  1  - 5  0   0   6  - 2

            1  -4  -4 -4  2

-----------------------          

        1  -4 -4  -4  2  0(remainder)

The second factor of the polynomial is `x^4 - 4x^3 - 4x^2-4x + 2`

This fourth-degree polynomial has no rational roots, but we can find approximate roots by using its graph:

We see from the graph that this function change its sign at values of x approximately equal to 0.5 and 5. This means the original fifth-degree polynomial changes sign at 0.5, 1 and 5

At x=0, the inequality `x^5 - 5x^4+6x - 2 <0 ` is true. Then, it is also true at x between 1 and 5.

So the approximate solution of the inequality is

`x<0.5` and `1<x<5`

 

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