Question

Let f(x) = 2x-5 and g(x) = 7-2x, Then $\left|f\right(x)+g(x\left)\right|=\left|f\right(x\left)\right|+\left|g\right(x\left)\right|$ if and only if

• A. $\frac{5}{2}<x<\frac{7}{2}$
• B. $x\le \frac{5}{2}orx\ge \frac{7}{2}$
• C. $x<\frac{5}{2}orx\ge \frac{7}{2}$
• D. $\frac{5}{2}\le x\le \frac{7}{2}$

Answer 1

The correct option is D

$\frac{5}{2}\le x\le \frac{7}{2}$

l f(x) + g(x) l = l f(x) l + l g(x) l
We know that la + bl = lal + lbl, if and only if both a and b are positive or negative simultaneously.
Best way to solve this question is by checking for different values of x.
For x = 5,
f(5) = 2(5) – 5 = 10 and g(5) = 7 – 2(5) = -3
Since, f(x) is positive and g(x) is negative. So this value is not possible. Hence, option (b) and (c) are eliminated.
For x = 5/2,
f(5/2) = 2(5/2) – 5 = 0 and g(x) = 7 – 2(5/2) = 2.
Since, f(x) and g(x) both have same sign (zero considered positive). Hence at x = 5/2, equation hold true.
So, option (a) is eliminated and hence correct answer is option (d).