Tim Burnham

Simplify the function 𝑛(𝑥) = (9𝑥 + 72)/(𝑥 + 1) ÷ (9𝑥 + 72)/(5𝑥 + 5).

Simplify the function π of π₯ is equal to nine π₯ plus 72 over π₯ plus one divided by nine π₯ plus 72 over five π₯ plus five.

Okay. So this is a dividing fractions question. And when dividing fractions, we actually flip the second fraction and turn it into a multiplication. So we can say π of π₯ is equal to nine π₯ plus 72 over π₯ plus one times five π₯ plus five over nine π₯ plus 72. Now I like to put parentheses around each of my numerators and denominators to just identify the fact that theyβre separate terms. And then we can see that in the numerator of the first fraction, weβve got nine π₯ plus 72. In the denominator of the second fraction, weβve got nine π₯ plus 72. So in multiplication of fractions, we multiply the numerators together, we multiply the denominators together. I can do some cross canceling here. So if I divide nine π₯ plus 72 by nine π₯ plus 72, I get one. If I divide nine π₯ plus 72 by nine π₯ plus 72, I also get one.

So I can now say that π of π₯ is equal to one over π₯ plus one times five π₯ plus five over one. And now I can see that I can factor this numerator here. Theyβve both got five as a common factor. So Iβm gonna take that out of that numerator. And five times π₯ is five π₯, and five times positive one is positive five. So now I can see that Iβve got a common factor of π₯ plus one on the top and on the bottom of those fractions. π₯ plus one divided by π₯ plus one is just one, and π₯ plus one divided by π₯ plus one is just one. So Iβve now got one times five times one on the numerator and one times one on the denominator. Well that simplifies to five over one or just five.

So weβve simplified that function down to π of π₯ is equal to five.

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