Certainly! Let's solve the equation [tex]\( 4 = \sqrt{\frac{c x + 1}{d x - 1}} \)[/tex] for [tex]\( x \)[/tex].
Step 1: Begin by eliminating the square root. To do this, square both sides of the equation:
[tex]\[
4^2 = \left(\sqrt{\frac{c x + 1}{d x - 1}}\right)^2
\][/tex]
This simplifies to:
[tex]\[
16 = \frac{c x + 1}{d x - 1}
\][/tex]
Step 2: Rearrange the equation to isolate [tex]\( c x + 1 \)[/tex] on one side. Multiply both sides by [tex]\( d x - 1 \)[/tex]:
[tex]\[
16 (d x - 1) = c x + 1
\][/tex]
Step 3: Expand and simplify the equation:
[tex]\[
16 d x - 16 = c x + 1
\][/tex]
Step 4: Gather all the [tex]\( x \)[/tex]-terms on one side and the constant terms on the other side:
[tex]\[
16 d x - c x = 1 + 16
\][/tex]
Step 5: Factor out [tex]\( x \)[/tex] on the left-hand side:
[tex]\[
x (16 d - c) = 17
\][/tex]
Step 6: Solve for [tex]\( x \)[/tex] by dividing both sides by [tex]\( 16 d - c \)[/tex]:
[tex]\[
x = \frac{17}{16 d - c}
\][/tex]
Thus, the solution in terms of [tex]\( c \)[/tex] and [tex]\( d \)[/tex] is:
[tex]\[
x = \frac{17}{16 d - c}
\][/tex]
