Solve the problem.
-For the function $f ( x ) = 5 \sqrt { x - 3 } - 5 \sqrt { 2 x - 4 }$, find all values $x$ for which $f ( x ) = 5$.
A) 5
B) 4
C) $- 5$
D) $\varnothing$

Question

Quizplus · Accepted Answer

The equation $f(x) = 5$ leads to $5 \sqrt{x - 3} - 5 \sqrt{2x - 4} = 5$. Simplifying this equation involves dealing with square roots and their domains. The domain restrictions for $x$ in the square roots $\sqrt{x - 3}$ and $\sqrt{2x - 4}$ are $x \geq 3$ and $x \geq 2$, respectively, ensuring real values. However, attempting to isolate $x$ in the given equation after moving terms or squaring both sides to eliminate the square roots does not yield a solution that satisfies both the original equation and the domain restrictions. This indicates that there is no real value of $x$ that satisfies the equation $f(x) = 5$ under the given function definition, leading to the conclusion that there are no solutions, represented by the empty set $\varnothing$.

Solve the Problem f(x)=5x−3−52x−4f ( x ) = 5 \sqrt { x - 3 } - 5 \sqrt { 2 x - 4 }f(x)=5x−3​−52x−4​

Solve the Problem $f ( x ) = 5 \sqrt { x - 3 } - 5 \sqrt { 2 x - 4 }$