Question

Sum of all the roots of the equation $||2x-3|-4| = 2$ is

• A. 8
• B. 0
• C. 6
• D. 9

Hint:

When solving equations with nested absolute values, like $| |ax+b| - c | = d$, work from the outside in. First solve $|Y| = d$ to get $Y=d$ or $Y=-d$, where $Y = |ax+b|-c$. Then solve each of these resulting equations for $x$.

Answer 1

The Correct Option is C

We are given the equation $||2x-3|-4| = 2$.
To solve this, we remove the outermost absolute value. This gives two possibilities.
Case 1: $|2x-3|-4 = 2$.
$|2x-3| = 6$.
This leads to two sub-cases:
Sub-case 1a: $2x-3 = 6 \Rightarrow 2x = 9 \Rightarrow x = 4.5$.
Sub-case 1b: $2x-3 = -6 \Rightarrow 2x = -3 \Rightarrow x = -1.5$.
Case 2: $|2x-3|-4 = -2$.
$|2x-3| = 2$.
This leads to two more sub-cases:
Sub-case 2a: $2x-3 = 2 \Rightarrow 2x = 5 \Rightarrow x = 2.5$.
Sub-case 2b: $2x-3 = -2 \Rightarrow 2x = 1 \Rightarrow x = 0.5$.
The roots of the equation are $\{4.5, -1.5, 2.5, 0.5\}$.
The sum of all the roots is $4.5 + (-1.5) + 2.5 + 0.5$.
Sum = $3.0 + 3.0 = 6$.