Let us split this into two cases. Case 1, when x is greater than 0 and Case 2, when x is lesser than 0.
Case 1
x > 0. Now, |x| = x
x^{2} – 7x – 18 = 0
(x – 9) (x + 2) = 0
x is either –2 or +9.
Case 2
x < 0. Now, |x| = –x
x^{2} + 7x – 18 = 0
(x + 9) (x – 2) = 0
x is either –9 or +2.
However, in accordance with the initial assumption that x < 0, x can only be –9 (cannot be +2).
Hence, this equation has two roots: –9 and +9.
Alternatively, we can treat this as a quadratic in |x|, the equation can be written as |x|2 – 7 |x| – 18 = 0.
Or, (|x| – 9) (|x| + 2) = 0
|x| = 9 or –2. |x| cannot be –2.
|x| = 9, x = 9 or –9.
Correct Answer: There are 2 solutions.
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