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
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?
Please be complete. Thank you!
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) $