Find the root of the equation 1/(4x - 5) = 1/11.

Step-by-step solution

1. 1

   State the domain restriction first. The left side is undefined when its denominator vanishes:

   $4x-5\ne 0\ \Longrightarrow\ x\ne\frac{5}{4}$

   Any candidate equal to $\tfrac54$ would have to be discarded at the end.

2. 2

   Equate the denominators. Both sides are reciprocals with numerator $1$, and $\dfrac1A=\dfrac1B$ with $A,B\ne 0$ forces $A=B$:

   $4x-5=11$

   (Cross-multiplying gives the identical equation $11=4x-5$.)

3. 3

   Solve the resulting linear equation. Add $5$ to both sides:

   $4x=16$

4. 4

   Divide by 4.

   $x=4$

5. 5

   Confirm the root is admissible and check. $4\ne\tfrac54$, so the domain restriction is satisfied. Substituting: $4(4)-5=11$, and $\dfrac{1}{11}=\dfrac{1}{11}\ \checkmark$