Question

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

Answer 1

i)

a)

$x<-1:$

$$-x-x-1<2$$

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

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

b)

$-1\leq x<0:$

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

$$2>1$$

$x\in[-1, 0)$

c)

$x\geq 0$

$$x+x+1<2$$

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

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

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

ii)

$$|x^2-1|\leq3$$

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

$$-2\leq x^2\leq4$$

$$-2\leq x\leq 2$$

Answer: $[-2, 2]$