How do you solve and write the following in interval notation: #(4-x) /( x-8)=0#?

Question

How do you solve and write the following in interval notation: #(4-x) /( x-8)>=0#?

Genesis Allen · Accepted Answer

The answer is #x in [ 4, -8 [#

Let #f(x)=(4-x)/(x-8)#

The domain of #f(x)# is #D_f(x)=RR-{8}#

Let us construct the sign chart.

#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaa)##4##color(white)(aaaaaaaa)##8##color(white)(aaaaaaa)##+oo#

#color(white)(aaaa)##4-x##color(white)(aaaa)##+##color(white)(aaaa)##-##color(white)(aaaa)##color(red)(||)##color(white)(aaaa)##-#

#color(white)(aaaa)##x-8##color(white)(aaaa)##-##color(white)(aaaa)##-##color(white)(aaaa)##color(red)(||)##color(white)(aaaa)##+#

#color(white)(aaaa)##f(x)##color(white)(aaaaa)##-##color(white)(aaaa)##+##color(white)(aaaa)##color(red)(||)##color(white)(aaaa)##-#

Consequently,

#f(x>=0)# when #x in [ 4, -8 [#

Eva Abramson · Answer

undefined To solve the inequality $\frac{{4-x}}{{x-8}} \geq 0$, you would find the critical points where the expression equals zero and where it is undefined. The critical points are $x = 4$ and $x = 8$. Then, you would test the intervals between these critical points to determine where the expression is positive or negative. Finally, you would express the solution in interval notation. The solution in interval notation is $[- \infty, 4] \cup (8, \infty)$.

How do you solve and write the following in interval notation: #(4-x) /( x-8)&gt;=0#?

How do you solve and write the following in interval notation: #(4-x) /( x-8)>=0#?