Not The Smallest

Let's compare the expressions pairwise (we do not allow equality here, because this would mean, that at least two of the expressions are equal, therefore there wouldn't be a smallest one).

$\text {a) } -2x + 2 < x + 5 \iff -3 < 3x \iff -1 < x$

$\text {b) } -2x + 2 < 3x + 2 \iff 0 < 5x \iff 0 < x$

$\text {c) } \hphantom { abcde } x + 5 < 3x + 2 \iff 3 < 2x \iff 1.5 < x$

Now, we can use these results to investigate the following cases:

$\textbf {Case 1: } \, \mathbf {x < -1} \hphantom {abcdefg} \text {We get: } \hphantom {abcde} 3x + 2 < x + 5 < - 2x + 2$

$\textbf {Case 2: } \, \mathbf {-1 < x < 0} \hphantom {abc} \text {We get: } \hphantom {abcde} 3x + 2 < -2x + 2 < x + 5$

$\textbf {Case 3: } \, \mathbf {0 < x < 1.5} \hphantom {abci} \text {We get: } \hphantom {abc} -2x + 2 < 3x + 2 < x + 5$

$\textbf {Case 4: } \, \mathbf {1.5 < x } \hphantom {abcdefgi} \text {We get: } \hphantom {abc} -2x + 2 < x + 5 < 3x + 2$

As we can see from these cases, whenever we have a smallest one amongst these three expressions, it cannot be:

$\boxed { x + 5 }$

Since these expressions seem easy to graph, we think of it like this: Since y=-2x+2 is the only negative one, at when x approaches infinity, it becomes the lowest. Since y=3x+2 is the steepest positive function, when x approaches negative infinity, it becomes the lowest. At x=0, the two above functions intersect with y=x+5 above them. Therefore, it is above at least one of the functions at all times. Therefore, x+5 never is the lowest.

Taking $x = 100$ and $x = -100$ shows that $-2x+2$ and $3x + 2$ can be the smallest number.

Now, we try to figure out the conditions for $x + 5$ to be the smallest : $x + 5 \leq 3x + 2 \implies x \geq 1.5 \\ x + 5 \leq -2x+2 \implies x \leq -1$

Hence, we see that $x+5$ can never be the smallest number.

We can also say this by the analogy that a negative value of x will make the third expressions more negative and a positive value can make them first expression more negative i.e. smaller . But any real value of x will directly influence the second expression. However a lot of thanks to Calvin sir , this is a GREAT Question...:):)