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[tex]\[
4 = \sqrt{\frac{c x + 1}{d x - 1}}
\][/tex]

Find [tex]\( x \)[/tex] in terms of [tex]\( c \)[/tex] and [tex]\( d \)[/tex].



Answer :

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]

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