Find the root of the equation 1/(3x + 11) = 1/(9x + 5).

Step-by-step solution

1. 1

   Use the rule for equal reciprocals instead of cross-multiplying blindly. If $\dfrac{1}{a}=\dfrac{1}{b}$ with $a\neq 0$ and $b\neq 0$, then taking reciprocals of both sides gives $a=b$. That turns a rational equation into a linear one in a single step:

   $3x+11=9x+5$

2. 2

   Note the excluded values before you solve. The two denominators must never be zero, so $x\neq -\dfrac{11}{3}$ and $x\neq -\dfrac{5}{9}$. Any root that lands on one of these is extraneous and must be discarded.

3. 3

   Solve the linear equation. Move the variable terms to one side and the constants to the other:

   $11-5=9x-3x$

   $6=6x$

   $x=1$

4. 4

   Check that $x=1$ is not an excluded value. Substitute into each denominator:

   $3(1)+11=14\neq 0,\qquad 9(1)+5=14\neq 0$

   Both denominators are $14$, so the root is legitimate.

5. 5

   Verify the original equation numerically.

   $\frac{1}{14}=\frac{1}{14}\quad\checkmark$

   The equality holds exactly, so $x=1$ is the only root.