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How do you solve #1/2+1/(2x) = (x^2-7x+10)/(4x)#?

Answer 1

I have taken you to a point where you can take over

#color(blue)("Preamble")#

The letters in algebra represent numbers. Consequently the rules that apply to numbers also apply to the letters (variables).

'Force' the denominators to be the same. Once this is the case we may directly compare the numerators and forget about the denominators. So by example:

Suppose we had: #z/2-3/2=56/2#

As all the denominators are 2 then it is equally true that

#z-3=56# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #color(blue)("Answering the question")# Multiply by 1 and you do not change the intrinsic value. However, 1 comes in many forms.

#color(green)([1/2color(red)(xx1)]+[1/(2x)color(red)(xx1)]" " =" "(x^2-7x+10)/(4x)#

#color(green)([1/2color(red)(xx(2x)/(2x))]+[1/(2x)color(red)(xx2/2)]" " =" "(x^2-7x+10)/(4x)#

#color(green)([(2x)/(4x)]+[2/(4x) ] =(x^2-7x+10)/(4x)#

So now it is equally true that:

#2x+2=x^2-7x+10#

Now solve as a quadratic set to 0

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.