Answer:
[tex]\large\boxed{\dfrac{x+1}{x}=1+\dfrac{1}{x}}[/tex]
Step-by-step explanation:
[tex]\dfrac{x^2-1}{x^2-x}=(*)\\\\x^2-1=x^2-1^2\qquad\text{use}\ a^2-b^2=(a-b)(a+b)\\=(x-1)(x+1)\\\\x^2-x=(x)(x)-(x)(1)\qquad\text{use the distributive property}\\=x(x-1)\\\\(*)=\dfrac{(x-1)(x+1)}{x(x-1)}\qquad\text{cancel}\ (x-1)\\\\=\dfrac{x+1}{x}=\dfrac{x}{x}+\dfrac{1}{x}=1+\dfrac{1}{x}[/tex]
The expression x^2-1 / x^2-x in its simplest form is 1 + 1/x.
The given expression can be simplified by eliminating the common terms from the denominator and numerator. By doing this, the expression can be simplified.
The given expression is:
[tex]\frac{x^{2}-1 }{x^{2}-x}[/tex]
It can be simplified as shown below:
[tex]\frac{x^{2}-1 }{x^{2}-x} = \frac{(x+1)(x-1 )}{x(x-1)}[/tex]
The numerator is split using the identity :
[tex]a^{2} -b^{2} =(a+b)(a-b)[/tex]
It can be further simplified as follows:
[tex]\frac{x^{2}-1 }{x^{2}-x} = \frac{(x+1)(x-1 )}{x(x-1)}\\= \frac{x+1}{x} \\=\frac{x}{x} +\frac{1}{x} \\= 1+\frac{1}{x}[/tex]
We have simplified the given expression into 1 + 1/x.
Thus, the expression x^2-1 / x^2-x in its simplest form is 1 + 1/x.
Learn more about simplifying here: https://brainly.com/question/723406
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