Question 261964
The reason why we're assuming that she has over $160 in medical bills is if she had less than $160 in bills, then she would be clearly paying more for plan B. For instance, if she has $159 in bills, then she pays almost $15 more for plan B.



Let x = amount Giselle has to pay (ie her bill)


Under plan A, she has to pay the first $140 and then 25% of the rest. So if we assume that she has over $160 in bills, this means that {{{x>160}}}. After she pays the initial $140, then she has 25% of {{{x-140}}} in bills left over. So this means that under plan A, the cost is {{{c=140+0.25(x-140)}}}



Similarly, under plan B, because "she would have to pay the first $160 of her bills and 20% of the rest", this means that the cost for plan B is {{{c=160+0.20(x-160)}}}




So to figure out when plan B will save her money, simply set the plan B expression less than the plan A expression



{{{Plan_B<Plan_A}}}



{{{160+0.20(x-160)<140+0.25(x-140)}}} Substitute the given cost equations.



{{{160+0.20x-32<140+0.25x-35}}} Distribute



{{{128+0.20x<105+0.25x}}} Combine like terms



{{{0.20x-0.25x<105-128}}} Subtract 0.25x from both sides. Subtract 128 from both sides. 



{{{-0.05x<-23}}} Combine like terms



{{{x>460}}} Divide both sides by -0.05. Remember that dividing both sides by a negative number will flip the inequality sign.



So if she has any bills over $460, then Plan B will cost less than Plan A. 


If you're skeptical, try some values of 'x' that are around $460. Try x=400, x=450, x=500 (and maybe more) and you'll notice that plan B will become cheaper after you pass x=450.


Note: the plans cost the same when the bill is $460. This is cross over point when the plans switch in cost.