Solving Rational Inequalities on a Graphing Calculator

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Key Questions

  • 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:

    enter image source here


    I hope that this was helpful.

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