Solve the inequality: (4x-4)/(x^(2)-1)\ \textgreater \ 1

Question

Solve the inequality:

`(4x-4)/(x^(2)-1)\ \textgreater \ 1`

Answer 1

Answer: To solve the inequality:

`(4x-4)/(x^(2)-1)\ \textgreater \ 1`

We can begin by multiplying both sides by
`x^2-1`:

tex\ >\ (x^{2}-1)[/tex]

Expanding the right-hand side:

`4x-4\ >\ x^(2)-1`

Rearranging:

`x^(2)-4x+3\ <\ 0`

Now we can factor the left-hand side:

tex(x-3)\ <\ 0[/tex]

To solve for the inequality, we need to find the values of x that make the expression on the left-hand side less than zero. We can do this by analyzing the sign of the expression for x-values in the intervals between its roots, which are x=1 and x=3.

If we test x=0, for example, we get:

tex(0-3)\ <\ 0[/tex]

Which is true, so the interval [1, 3] must be excluded from the solution set. The solution is the union of the intervals that make the expression less than zero, which in this case is:

`(-\infty, 1)\ \cup\ (3, \infty)`

Therefore, the solution to the inequality is
`x\ \in\ (-\infty, 1)\ \cup\ (3, \infty)`.

Explanation: