Elias Heath

2022-09-27

$\text{}f(x)={\mathrm{log}}_{x-2}\left(\frac{2x+3}{7-x}\right)1$

Which is the solution of $\text{}f(x)1$ ? I did:

$$\text{}\frac{2x+3}{7-x}0$$

$\text{}x7$ results in $\text{}x\frac{-3}{2}$ whereas $\text{}x7$ results in $\text{}x\frac{-3}{2}$ so x=($\text{}\frac{-3}{2}$,7) and also $\text{}x-20$ so $\text{}x2$

$$\text{}x=(2,7)-\{3\}$$

So after finding out where f is defined, I did this:

$${\mathrm{log}}_{x-2}\left(\frac{2x+3}{7-x}\right)<\mathrm{log}{\phantom{\rule{mediummathspace}{0ex}}}_{x-2}(x-2)$$

$$\text{}\frac{2x+3}{7-x}x-2$$

$$\text{}{x}^{2}-7x+170$$

And I figured I must have done a mistake somewhere as $\text{}d0$ and $\text{}{x}^{2}$'s coefficient is $\text{}0$ so $\text{}f$ would be $\text{}0$. Could you let me know where is my mistake? The correct answer is $\text{}x=(2,3)$

Which is the solution of $\text{}f(x)1$ ? I did:

$$\text{}\frac{2x+3}{7-x}0$$

$\text{}x7$ results in $\text{}x\frac{-3}{2}$ whereas $\text{}x7$ results in $\text{}x\frac{-3}{2}$ so x=($\text{}\frac{-3}{2}$,7) and also $\text{}x-20$ so $\text{}x2$

$$\text{}x=(2,7)-\{3\}$$

So after finding out where f is defined, I did this:

$${\mathrm{log}}_{x-2}\left(\frac{2x+3}{7-x}\right)<\mathrm{log}{\phantom{\rule{mediummathspace}{0ex}}}_{x-2}(x-2)$$

$$\text{}\frac{2x+3}{7-x}x-2$$

$$\text{}{x}^{2}-7x+170$$

And I figured I must have done a mistake somewhere as $\text{}d0$ and $\text{}{x}^{2}$'s coefficient is $\text{}0$ so $\text{}f$ would be $\text{}0$. Could you let me know where is my mistake? The correct answer is $\text{}x=(2,3)$

Emaidedip6g

Beginner2022-09-28Added 11 answers

$${\mathrm{log}}_{(x-2)}\left(\frac{2x+3}{7-x}\right)<1$$

Hint

$1)$$x-2>0$ and $x-2\ne 1$

$2)$ $\frac{2x+3}{7-x}>0$

$2)$ If $x-2>1$ then

$$\frac{2x+3}{7-x}<x-2$$

$3)$ If $0<x-2<1$ then

$$\frac{2x+3}{7-x}>x-2$$

Now you can solve it.

Hint

$1)$$x-2>0$ and $x-2\ne 1$

$2)$ $\frac{2x+3}{7-x}>0$

$2)$ If $x-2>1$ then

$$\frac{2x+3}{7-x}<x-2$$

$3)$ If $0<x-2<1$ then

$$\frac{2x+3}{7-x}>x-2$$

Now you can solve it.

tarjetaroja2t

Beginner2022-09-29Added 2 answers

The domain gives $2<x<7$, $x\ne 3$ and since $\frac{2x+3}{7-x}\ne x-2$

we get the answer without any cases immediately by the intervals method:

$$(2,3).$$

Done!

we get the answer without any cases immediately by the intervals method:

$$(2,3).$$

Done!

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