Решите уравнение |х ─ 1|+|х ─ 3|+|х ─ 7| = 6.. Created by GUNKINA. algebra-ru

Решите уравнение |х ─ 1|+|х ─ 3|+|х ─ 7| = 6...

GUNKINA @GUNKINA

July 2022 1 64 Report

Решите уравнение |х ─ 1|+|х ─ 3|+|х ─ 7| = 6.

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|x-1|+|x-3|+|x-7|=6
x-1=0 x-3=0 x-7=0
x=1 x=3 x=7 __________1__________3________7__________

1) x≤1 -(x-1)-(x-3)-(x-7)=6
-x+1-x+3-x+7=6
-3x+11=6
-3x=6-11
-3x=-5
x=5/3 (>1)
x=5/3∉(-∞;1)
На данном интервале решений нет
2) 1<x≤3 +(x-1)-(x-3)-(x-7)=6
x-1-x+3-x+7=6
-x+9=6
-x=6-9
-x=-3
x=3 ∈(1;3]
x=3 - решение уравнения
3) 3>x≤7 +(x-1)+(x-3)-(x-7)=6
x-1+x-3-x+7=6
x+3=6
x=6-3
x=3∉[3;7)
На данном интервале решений нет
4) x>7 +(x-1)+(x-3)+(x-7)=6
x-1+x-3+x-7=6
3x-11=6
3x=6+11
3x=17
x=17/3
x= $5 \frac{2}{3}$ (<7)
х= $5 \frac{2}{3}$ ∉(7;+∞)
На данном интервале решений нет
Ответ: 3

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