Example  1
When applying for a passport, a person needs to supply two identical passport photos. The photo needs to be \(70 \rm\thinspace {mm}\) by \(50 \rm\thinspace {mm}\). In the photo, the face of the person needs to be \(33.5 \pm 2.5 \rm \thinspace {mm}\) tall.

  1. Model the acceptable maximum and minimum face heights, \(h\), using an absolute value equation.

    The difference between \(33.5 \thinspace \rm {mm}\) and the face height cannot exceed \(2.5 \thinspace \rm {mm}\), so the situation can be modelled by the equation \(\left|33.5 - h\right| = 2.5\).
  2. Solve the equation.

    Case 1: \(33.5 - h \ge 0\)

    \(\begin{align}
     33.5 - h &\ge 0 \\
     33.5 &\ge h \end{align}\)

    This case occurs when \(33.5 \ge h\).
    \(\begin{align}
     \left| {33.5 - h} \right| &= 2.5 \\
     33.5 - h &= 2.5 \\
      - h &= - 31 \\
     h &= 31 \end{align}\)

    A height of \(31\) satisfies \(33.5 \ge h\), so it is a solution to the equation.

    Case 2: \(33.5 - h \lt 0\)

    \(\begin{align}
     33.5 - h &\lt 0 \\
     33.5 &\lt h \end{align}\)

    This case occurs when \(33.5 \lt h\).
    \(\begin{align}
     \left| {33.5 - h} \right| &= 2.5 \\
     -(33.5 - h) &= 2.5 \\
      - 33.5 + h &= 2.5 \\
     h &= 36 \end{align}\)

    A height of \(36\) satisfies \(33.5 \lt h\), so it is a solution to the equation.
  3. Explain the solution using a number line.

    The minimum height occurs at \(31 \thinspace \rm{mm}\) and the maximum height occurs at \(36 \thinspace \rm{mm}\).