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

We have $f(x) = 2x - 5$ and $g(x) = 7 - 2x$. We need to find when $|f(x) + g(x)| = |f(x)| + |g(x)|$.

---

First, let's find $f(x) + g(x)$:

$f(x) + g(x) = (2x - 5) + (7 - 2x) = 2x - 5 + 7 - 2x = 2$

The $x$ terms cancel out completely!

---

Since $f(x) + g(x) = 2$, our equation becomes:

$|2| = |f(x)| + |g(x)|$

$2 = |2x - 5| + |7 - 2x|$

---

For absolute values, $|A| + |B| = |A + B|$ only when $A$ and $B$ have the same sign.

Since $f(x) + g(x) = 2 > 0$, we need both $f(x) \geq 0$ and $g(x) \geq 0$.

---

For $f(x) \geq 0$:

$2x - 5 \geq 0$

$2x \geq 5$

$x \geq \dfrac{5}{2}$

---

For $g(x) \geq 0$:

$7 - 2x \geq 0$

$7 \geq 2x$

$x \leq \dfrac{7}{2}$

---

We need both conditions to be true simultaneously:

$x \geq \dfrac{5}{2}$ AND $x \leq \dfrac{7}{2}$

Therefore: $\dfrac{5}{2} \leq x \leq \dfrac{7}{2}$