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Number of real roots of an equation with modulus terms (split at the critical points)

This type of question, asked 10 times: 2021 Paper 2, 2021 Paper 15, 15 Apr 2023, 30 Jan 2024, 5 Apr 2024, 8 Apr 2024, 24 Jan 2025, 7 Apr 2025, 22 Jan 2026 S1, 24 Jan 2026 S1. All 11 questions below, oldest first.

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  1. Question 1

    2021 Paper 2

    81
    The number of the real roots of the equation (x+1)2+∣x−5∣=274(x+ 1)^{2}+ |x- 5| =\frac{27}{4} is

  2. Question 2

    2021 Paper 15

    43
    The number of real solutions of x2−∣x∣−12=0x^{2}- |x| - 12 = 0 is

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  3. Question 3

    15 Apr 2023

    67
    The number of real roots of the equation x∣x∣−5∣x+2∣+6=0x|x| - 5|x + 2| + 6 = 0, is

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  4. Question 4

    30 Jan 2024

    178
    The number of real solutions of the equation x(x2+3∣x∣+5∣x−1∣+6∣x−2∣)=0x\left( x^{2}+ 3|x| + 5|x- 1| + 6|x- 2| \right)= 0 is

  5. Question 5

    5 Apr 2024

    89
    The number of distinct real roots of the equation ∣x∣∣x+2∣−5∣x+1∣−1=0|x||x+ 2| - 5|x+ 1| - 1 = 0 is

  6. Question 6

    5 Apr 2024

    180
    The number of real solutions of the equation x∣x+5∣+2∣x+7∣−2=0x|x+ 5| + 2|x+ 7| - 2 = 0 is

  7. Question 7

    8 Apr 2024

    176
    The number of distinct real roots of the equation ∣x+1∣∣x+3∣−4∣x+2∣+5=0|x + 1||x + 3| - 4|x + 2| + 5 = 0, is

  8. Question 8

    24 Jan 2025

    143
    The number of real solution(s) of the equation x2+3x+2=min⁡{∣x−3∣,∣x+2∣}x^{2}+ 3x+ 2 = \min\{|x- 3|,|x+ 2|\} is :

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  9. Question 9

    7 Apr 2025

    142
    The number of real roots of the equation x∣x−2∣+3∣x−3∣+1=0x|x - 2| + 3|x - 3| + 1 = 0 is :

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  10. Question 10

    22 Jan 2026 S1

    58
    The number of distinct real solutions of the equation x∣x+4∣+3∣x+2∣+10=0x|x + 4| + 3|x + 2| + 10 = 0 is

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  11. Question 11

    24 Jan 2026 S1

    70
    The number of the real solutions of the equation : x∣x+3∣+∣x−1∣−2=0x|x + 3| + |x - 1| - 2 = 0 is

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