Provide an appropriate response.
-Graph the functions $f ( x ) = \frac { x } { 2 }$ and $g ( x ) = 3 + \frac { 8 } { x }$ together to identify the values of $x$ for which $\frac { x } { 2 } 3 + \frac { 8 } { x }$. Confirm your findings algebraically.
A) $( - 2,0 ) \cup ( 8 , \infty )$
B) $( 8 , \infty )$
C) $( - \infty , - 2 ) \cup ( 0,8 )$
D) $( - 2,8 )$

Question

Provide an appropriate response.
-Graph the functions $f ( x ) = \frac { x } { 2 }$ and $g ( x ) = 3 + \frac { 8 } { x }$ together to identify the values of $x$ for which $\frac { x } { 2 } > 3 + \frac { 8 } { x }$. Confirm your findings algebraically.
A) $( - 2,0 ) \cup ( 8 , \infty )$
B) $( 8 , \infty )$
C) $( - \infty , - 2 ) \cup ( 0,8 )$
D) $( - 2,8 )$

Quizplus · Accepted Answer

To solve the inequality $\frac{x}{2} > 3 + \frac{8}{x}$ algebraically, we first bring all terms to one side: $\frac{x}{2} - 3 - \frac{8}{x} > 0$. Multiplying through by $2x$ (assuming $x \neq 0$) to clear the fraction gives $x^2 - 6x - 16 > 0$. Factoring or using the quadratic formula gives roots at $x = -2$ and $x = 8$. Testing intervals gives us that the inequality is satisfied for $x \in ( - 2,0 ) \cup ( 8 , \infty )$.

Provide an Appropriate Response f(x)=x2f ( x ) = \frac { x } { 2 }f(x)=2x​

Provide an Appropriate Response $f ( x ) = \frac { x } { 2 }$