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Find the number of roots of the equation 3|x| - |2 - |x|| = 1.

Mathematics

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Part a(i)· 2 steps

Find the number of roots of the equation 3∣x∣−∣2−∣x∣∣=13|x| - |2 - |x|| = 1.

  1. Substitute y=∣x∣y = |x|.

    Since y=∣x∣y = |x|, we know that y≥0y \ge 0. The equation becomes:

    3y−∣2−y∣=13y - |2 - y| = 1

  2. Solve for yy by considering cases for the absolute value ∣2−y∣|2 - y|.

    Case 1: 2−y≥0  ⟹  y≤22 - y \ge 0 \implies y \le 2.
    In this case, ∣2−y∣=2−y|2 - y| = 2 - y.

    3y−(2−y)=13y - (2 - y) = 1
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