Question

For all real numbers x the condition |3x - 20| + |3x - 40| = 20 necessarily holds if

• A. $6 < x < 11$
• B. $7 < x < 12$
• C. $10 < x < 15$
• D. $9 < x < 14$

Answer 1

Answer: (B)

$7 < x < 12$

Answer 2

The minimum value of the modulus function is zero, which means the minimum value for each of the two modulus functions occurs at $x =\frac{ 20}{3}$ and $x = \frac{40}{3}$, respectively.

For values of x less than $\frac{ 20}{3}$, both functions increase. Similarly, for x greater than $\frac{40}{3}$, the sum of the functions increases. In the range between $\frac{ 20}{3}$ and $\frac{40}{3}$, the values of the functions remain the same, as one function's increase matches the other's decrease.

As a result, the sum of the modulus functions remains at 20 within the range of values $[\frac{ 20}{3}, \frac{40}{3}]$, which can be expressed as $[6.66, 13.33].$

This condition is satisfied by the option: $7 < x < 12.$