Solving Rational Inequalities on a Graphing Calculator
Key Questions
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Let us solve the following rational inequality.
#f(x)={x+1}/{x^2+x-6} 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+x-6=(x+3)(x-2)=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.