simplify u^2+3u/u^2-9A.u/u-3, =/ -3, and u=/3B. u/u-3, u=/-3
Question
Answer:
The correct answer is: Answer choice: [A]:__________________________________________________________
→ "[tex] \frac{u}{u-3} [/tex] " ; " { u [tex] \neq [/tex] ± 3 } " ;
→ or, write as: " u / (u − 3) " ; {" u ≠ 3 "} AND: {" u ≠ -3 "} ;
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Explanation:
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We are asked to simplify:
[tex] \frac{(u^2+3u)}{(u^2-9)} [/tex] ;
Note that the "numerator" —which is: "(u² + 3u)" — can be factored into:
→ " u(u + 3) " ;
And that the "denominator" —which is: "(u² − 9)" — can be factored into:
→ "(u − 3) (u + 3)" ;
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Let us rewrite as:
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→ [tex] \frac{u(u+3)}{(u-3)(u+3)} [/tex] ;
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→ We can simplify by "canceling out" BOTH the "(u + 3)" values; in BOTH the "numerator" AND the "denominator" ; since:
" [tex] \frac{(u+3)}{(u+3)} = 1 [/tex] " ;
→ And we have:
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→ " [tex] \frac{u}{u-3} [/tex] " ; that is: " u / (u − 3) " ; { u[tex] \neq 3[/tex] } .
and: { u[tex] \neq-3[/tex] } .
→ which is: "Answer choice: [A] " .
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NOTE: The "denominator" cannot equal "0" ; since one cannot "divide by "0" ;
and if the denominator is "(u − 3)" ; the denominator equals "0" when "u = -3" ; as such:
"u[tex] \neq [/tex]3" ;
→ Note: To solve: "u + 3 = 0" ;
Subtract "3" from each side of the equation;
→ " u + 3 − 3 = 0 − 3 " ;
→ u = -3 (when the "denominator" equals "0") ;
→ As such: " u [tex] \neq [/tex] -3 " ;
Furthermore, consider the initial (unsimplified) given expression:
→ [tex] \frac{(u^2+3u)}{(u^2-9)} [/tex] ;
Note: The denominator is: "(u² − 9)" .
The "denominator" cannot be "0" ; because one cannot "divide" by "0" ;
As such, solve for values of "u" when the "denominator" equals "0" ; that is:
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→ " u² − 9 = 0 " ;
→ Add "9" to each side of the equation ;
→ u² − 9 + 9 = 0 + 9 ;
→ u² = 9 ;
Take the square root of each side of the equation;
to isolate "u" on one side of the equation; & to solve for ALL VALUES of "u" ;
→ √(u²) = √9 ;
→ | u | = 3 ;
→ " u = 3" ; AND; "u = -3 " ;
We already have: "u = -3" (a value at which the "denominator equals "0") ;
We now have "u = 3" ; as a value at which the "denominator equals "0");
→ As such: " u[tex] \neq 3[/tex]" ; "u [tex] \neq [/tex] -3 " ;
or, write as: " { u [tex] \neq [/tex] ± 3 } " .
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solved
general
10 months ago
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