Example  1
The sum of the reciprocals of two consecutive odd integers is \(\frac{24}{143}\). Determine the two integers.


Step 1: Define the variables and expressions being used.

Let \(2x + 1\) represent the first odd integer, and let \(2x + 3\) represent the second consecutive odd integer.

Step 2: Write an equation to represent the problem.

\[\frac{1}{{2x + 1}} + \frac{1}{{2x + 3}} = \frac{{24}}{{143}}\]

Step 3: Identify any non-permissible values and the lowest common denominator.

NPVs:

\[\begin{align}
 2x + 1 &\ne 0 \\
 x &\ne -\frac{1}{2} \\
 \end{align}\]
\[\begin{align}
 2x + 3 &\ne 0 \\
 x &\ne -\frac{3}{2} \\
 \end{align}\]
LCD: \(143\left(2x + 1\right)\left(2x + 3\right)\)

Step 4: Solve for \(x\).

\[\begin{align}
 \left[ {\frac{1}{{{\color{red}\cancel{\color{#444}{2x + 1}}}}}} \right]\left( {143} \right){\color{red}\cancel{\color{#444}{\left(2x + 1\right)}}}\left( {2x + 3} \right) + \left[ {\frac{1}{{{\color{red}\cancel{\color{#444}{2x + 3}}}}}} \right]\left( {143} \right)\left( {2x + 1} \right){\color{red}\cancel{\color{#444}{\left( {2x + 3} \right)}}} &= \left[ {\frac{{24}}{{{\color{red}\cancel{\color{#444}{143}}}}}} \right]\left( {{\color{red}\cancel{\color{#444}{143}}}} \right)\left( {2x + 1} \right)\left( {2x + 3} \right) \\
 143\left( {2x + 3} \right) + 143\left( {2x + 1} \right) &= 24\left( {2x + 1} \right)\left( {2x + 3} \right) \\
 286x + 429 + 286x + 143 &= 24\left( {4x^2 + 8x + 3} \right) \\
 572x + 572 &= 96x^2 + 192x + 72 \\
 0 &= 96x^2 - 380x - 500 \\
 0 &= 4\left( {24x^2 - 95x - 125} \right) \\
 \end{align}\]

\[\begin{align}
 a &= 24,\thinspace b = - 95,\thinspace c = - 125 \\
 x &= \frac{{ - b \pm \sqrt {b^2 - 4ac} }}{{2a}} \\
 x &= \frac{{ - \left( { - 95} \right) \pm \sqrt {\left( { - 95} \right)^2 - 4\left( {24} \right)\left( { - 125} \right)} }}{{2\left( {24} \right)}} \\
 x &= \frac{{95 \pm \sqrt {21\thinspace 025} }}{{48}} \\
 x &= \frac{{95 \pm 145}}{{48}} \\
 x &= 5, \thinspace -\frac{{25}}{{24}} \\
 \end{align}\]

Because the question states that the numbers are integers, the negative fraction can be ignored.

Step 5: Determine the value of the integers.

The two numbers are:

\(\begin{align}
 2x + 1 &= 2\left( 5 \right) + 1 \\
  &= 11 \\
  \\
 2x + 3 &= 2\left( 5 \right) + 3 \\
  &= 13 \\
 \end{align}\)

Verify:

Left Side Right Side
\[\begin{array}{r}
 \frac{1}{{11}} + \frac{1}{{13}} \\
 \frac{{13}}{{143}} + \frac{{11}}{{143}} \\
 \frac{{24}}{{143}} \\
 \end{array}\]
 \[\frac{24}{143}\]
      LS = RS

The two integers are \(11\) and \(13\).