# Thread: [SOLVED] help with this inequality

1. ## [SOLVED] help with this inequality

Solve the following inequality. Write the answer in interval notation.

I got:

(-inifinity, 23/5)U(4,infinity)

&& got it wrong
does anyone know how to figure this out

2. How did you arrive at your answer? You added the 6 to the left-hand side, converted to the common denominator, combined the two fractions into one, found the zeroes of the numerator and denominator, tested the sign on each interval, and... then what?

3. huh?

i was left with

(-5x-23)/(x-4)

which was where i got my answer that is wrong

4. is it all real numbers?

5. no it is not all real numbers

because i kno by (-5x-23)/x-4

that the x in the denominator is 4.

but i am not sure if that fraction is right at all though

6. Originally Posted by lsnyder
no it is not all real numbers

because i kno by (-5x-23)/x-4

that the x in the denominator is 4.

but i am not sure if that fraction is right at all though
oops, isnt it
25/7<=x<4

x-1/x-4 <= -6
=x-1<= -6x+24
=7x-1<=24
=7x<=25
x<=25/7

and

x-1/x-4 <= -6
x-4<=-6x+24
7x<=28
x<=4
(i'm not sure why it's not equal to 4. When I worked it out on a calculator, it came up as x<4)

7. Hello, lsnyder!

Solve the following inequality. Write the answer in interval notation.
. . . $\displaystyle \frac{x-1}{x-4} \:\leq\:-6$

I got: .$\displaystyle \left(-\infty, \tfrac{23}{5}\right) \cup (4, \infty)$ . . . and got it wrong.

Does anyone know how to figure this out ?

After several false starts, I was forced to graph the function.

We have: .$\displaystyle y \:=\:\frac{x-1}{x-4}$

. . When is it below $\displaystyle y = -6$ ?
Code:
          |
|         :*
|         :
|         : *
|         :   *
|         :       *
|         :               *
- - - 1+ - - - - : - - - - - - - - -
*     |         :
*         :4
--------+---*-----+--------------------
|      *  :
|       o :
|         :
|        o:
|         :

The function has a vertical asymptote $\displaystyle x = 4$ and horizontal asymptote $\displaystyle y = 1$.

We see that for values of $\displaystyle x$ slightly below 4, the graph is below $\displaystyle y = -6$

If the function equals -6, we have: .$\displaystyle \frac{x-1}{x-4} \:=\:-6 \quad\Rightarrow\quad x \:=\:\frac{25}{7}$

Therefore, the interval is: .$\displaystyle \left[\frac{25}{7},\,4\right)$

8. why is it not equal to 4?