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Consider the two expressions $1$ and $\frac{2x+3}{2x+3}.$

a) Show that the two expressions represent equal numbers when $x=10.$

b) Explain why these two expressions do not represent equal numbers when $x=-\dfrac32.$

c) Show that these two expressions represent equal numbers for all $x$ other than $-\dfrac32.$

In parts (a) and (c), begin by explaining what your strategy for solving will be.

 May 5, 2024
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Haha, this is a very funny problem and shows the world of fractions. 

 

a) When x = 10, we have \(\frac{2(10)+3}{2(10)+3} = \frac{23}{23} = 1\). So yes, when x = 10, we have the first expression is equal to the second expression. 

 

b) Ok, so then why doesn't \(-\frac{3}{2}\) work? It looks like the numerator and denominator are equal, so it's always 1 right? 

 

Well, let's simplify and go from there. We have \(\frac{2(-2/3)+3}{2(-2/3)+3}=\frac{-3+3}{-3 + 3}=0/0\). WAIT a minute, the denominator can't be 0! That would make the value undefined, not 1!

 

c) Ok, so the reaon why the answer is not 1 in part (b) is because the denominator would become 0. 

 

However, as long as it doesn't equal 0, the numerator and denominator will always be equal, meaning the value of the fraction will always be 1!

 

I hope I answered your question!

 

Thanks!

 May 5, 2024

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