Triangle Inequality Equality Condition

Question

Let $f(x) = 2x – 5$ and $g(x) = 7 – 2x$. Then $|f(x) + g(x)| = |f(x)| + |g(x)|$ if and only if

Accepted Answer

D. The property $|A + B| = |A| + |B|$ holds if and only if $A$ and $B$ have the same sign, i.e., $A \cdot B \ge 0$. Here $A = f(x) = 2x - 5$ and $B = g(x) = 7 - 2x$. $$(2x - 5)(7 - 2x) \ge 0$$ $$-(2x - 5)(2x - 7) \ge 0$$ $$(2x - 5)(2x - 7) \le 0$$ Thus, $\frac{5}{2} \le x \le \frac{7}{2}$.

Answer

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

Answer

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

Answer

$x < \frac{5}{2}$ or $x \ge \frac{7}{2}$

Answer

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

Triangle Inequality Equality Condition

Explanation

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