First slide

Introduction to real valued functions

Question

The complete solution set of the equation $|x^{2} - 5 x + 6| + |x^{2} + 12 x + 27| = | 17 x + 21 |$ is

Moderate

A

$x \in [- 9, 3]$
B

$x \in [- 3, 2) \cup (2, 3]$
C

$x \in [- 9, - 3] \cup [2, 3]$
D

$x \in (- 2, 3)$

Solution

We have $|x^{2} - 5 x + 6| + |x^{2} + 12 x + 27| = | 17 x + 21 |$
or $|x^{2} - 5 x + 6| + |x^{2} + 12 x + 27|$
$= |(x^{2} + 12 x + 27) - (x^{2} - 5 x + 6)|$
$\begin{array}{l} \Rightarrow (x^{2} - 5 x + 6) (x^{2} + 12 x + 27) \leq 0 \\ \Rightarrow (x - 2) (x - 3) (x + 3) (x + 9) \leq 0 \end{array}$
From the sign scheme, $x \in [- 9, - 3] \cup [2, 3]$

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