Question

The number of solutions to the equation \(|x-1|+|x-3|=4\) is:

• A. 2
• B. 3
• C. 4
• D. Infinite

Hint:

Split absolute value equations into intervals.

Answer 1

The Correct Option is A

To find the number of solutions to the equation |x-1|+|x-3|=4, we need to consider the critical points of the expressions inside the absolute values, which are \( x = 1 \) and \( x = 3 \). These points divide the number line into three intervals:

1. \((-∞, 1)\)
2. \([1, 3]\)
3. \((3, ∞)\)

We will solve the equation in each interval separately:

1. For \(x \lt 1\):
   • Here, both \(x-1\) and \(x-3\) are negative.
   • Equation becomes: \(|x-1| + |x-3| = -(x-1) - (x-3) = -x + 1 - x + 3 = 4\).
   • Simplifying gives: \(-2x + 4 = 4\), which simplifies to \(x = 0\).
   • \(x = 0\) lies in the interval \((-∞, 1)\), so it is a valid solution.
2. For \(1 \leq x \leq 3\):
   • Here, \(x-1\) is non-negative and \(x-3\) is non-positive.
   • Equation becomes: \(|x-1| + |x-3| = (x-1) - (x-3) = 4\).
   • Simplifying gives: \(x - 1 - x + 3 = 4\), which reduces to \(2 = 4\).
   • This is a contradiction, so there are no solutions in this interval.
3. For \(x \gt 3\):
   • Here, both \(x-1\) and \(x-3\) are positive.
   • Equation becomes: \(|x-1| + |x-3| = (x-1) + (x-3) = 4\).
   • Simplifying gives: \(2x - 4 = 4\), which further simplifies to \(x = 4\).
   • \(x = 4\) lies in the interval \((3, ∞)\), so it is a valid solution.

Thus, the solutions to the equation are \(x = 0\) and \(x = 4\). Consequently, the number of solutions is 2.