Modulus ends at 1

Algebra Level 3

∣ ∣ ∣ ∣ ∣ x − 5 ∣ − 4 ∣ − 3 ∣ − 2 ∣ − 1 ∣ = 0 \LARGE \left| \left| \left| \left| \left| x-5 \right| -4 \right| -3 \right| -2 \right| -1 \right| = 0

How many solution we get for x x from this equation?

10 16 9 14 11 12

Md Mehedi Hasan by Md Mehedi Hasan , 22, Bangladesh

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1 solution

Md Mehedi Hasan
Dec 8, 2017

∣ ∣ ∣ ∣ ∣ x − 5 ∣ − 4 ∣ − 3 ∣ − 2 ∣ − 1 ∣ = 0 ∣ ∣ ∣ ∣ x − 5 ∣ − 4 ∣ − 3 ∣ − 2 ∣ = 1 ∣ ∣ ∣ x − 5 ∣ − 4 ∣ − 3 ∣ − 2 = ± 1 ∣ ∣ ∣ x − 5 ∣ − 4 ∣ − 3 ∣ = 2 ± 1 = 3 or 1 ∣ ∣ x − 5 ∣ − 4 ∣ − 3 = ± 3 , ± 1 ∣ ∣ x − 5 ∣ − 4 ∣ = 6 , 0 , 4 , 2 ∣ x − 5 ∣ − 4 = ± 6 , 0 , ± 4 , ± 2 ∣ x − 5 ∣ = 10 , − 2 , 4 , 8 , 0 , 6 , 2 x − 5 = ± 10 , ± 2 , ± 4 , ± 8 , 0 , ± 6 x = 15 , − 5 , 7 , 3 , 9 , 1 , 13 , − 3 , 5 , 11 , − 1 The total solution is: 11 \large{ \left| \left| \left| \left| \left| x-5 \right| -4 \right| -3 \right| -2 \right| -1 \right| = 0\\ \left| \left| \left| \left| x-5 \right| -4 \right| -3 \right| -2 \right| =1\\ \left| \left| \left| x-5 \right| -4 \right| -3 \right| -2=\pm1\\ \left| \left| \left| x-5 \right| -4 \right| -3 \right| =2\pm1=3\text{ or }1\\ \left| \left| x-5 \right| -4 \right| -3 =\pm3,\pm1\\ \left| \left| x-5 \right| -4 \right| =6,0,4,2\\ \left| x-5 \right| -4=\pm6,0,\pm4,\pm2\\ \left| x-5 \right| =10,-2,4,8,0,6,2\\ x-5 = \pm10,\pm2,\pm4,\pm8,0,\pm6\\ x=15,-5,7,3,9,1,13,-3,5,11,-1\\ \text{The total solution is: } \boxed{11}}

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