Question #278883

Find all x belongs to R that satisfy the following inequalities: i)|x|+|x+1|<2 ii)|x^2-1|<_3


1
Expert's answer
2021-12-13T16:28:05-0500

i)

a)

x<1:x<-1:


xx1<2-x-x-1<2

x>32x>-\dfrac{3}{2}

x(32,1)x\in(-\dfrac{3}{2}, -1)


b)

1x<0:-1\leq x<0:


x+x+1<2-x+x+1<2

2>12>1

x[1,0)x\in[-1, 0)


c)

x0x\geq 0


x+x+1<2x+x+1<2

x<12x<\dfrac{1}{2}

x[0,12)x\in[0, \dfrac{1}{2})

Answer: (32,12)(-\dfrac{3}{2}, \dfrac{1}{2})


ii)


x213|x^2-1|\leq3

3x213-3\leq x^2-1\leq 3

2x24-2\leq x^2\leq4

2x2-2\leq x\leq 2

Answer: [2,2][-2, 2]


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