Solving Rational Inequalities on a Graphing Calculator
Key Questions

Let us solve the following rational inequality.
#f(x)={x+1}/{x^2+x6} le 0# Set the numerator equal to zero, and solve for
#x# .#x+1=0 => x=1# (Note:
#f(1)=0# )Set the denominator equal to zero, and solve for
#x# .#x^2+x6=(x+3)(x2)=0 => x=3,2# (Note:
#f(3)# and#f(2)# are undefined.)Using
#x=3,1,2# above to split the number line into open intervals:#(infty,3), (3,1),(1,2)# , and#(2,infty)# Using sample numbers
#x=4,2,0,3# for each interval above, respectively, we can determine the sign of (LHS).#f(4)=2<0 => f(x)<0# on#(infty,3)# #f(2)=1/4>0 => f(x)>0# on#(3,1)# #f(0)=1/6<0 => f(x)<0# on#(1,2)# #f(3)=2/3>0 => f(x)>0# on#(2,infty)# Hence,
#f(x) le 0# on#(infty,3)cup[1,2)# .(Note:
#1# is included since#f(1)=0# .)The graph of
#y=f(x)# looks like:
I hope that this was helpful.