Inequalities Involving Modulus Function

Q. The solution set of the inequality $||x|-1<1-x,2\forall xϵR$ is equal to

1. $(0,\infty )$
2. $(-1,\infty )$
3. $(-1,1)$
4. $(-\infty ,0)$

Q.

Which of the following is the possible value of |x| if $|x{|}^2$ - 6|x| + 8 = 0

1. 2

2. 8

3. 1

4. 5

Q.

If $||x-2|-5|\le$ 10, then

1. -20 $\le$ x $\le$ 15

2. -5 $\le$ x $\le$ 27

3. -5 $\le$ x $\le$ 17

4. -13 $\le$ x $\le$ 17

Q.

Find the value of x if $x^2-4\left(x\right|+3=0$

1. 1

2. -1

3. 3

4. -3

Q.

$\text{Find the value of x if x if}|x+2|$= 3

1. 1

2. 5

3. -1

4. -5

Q.

Find the range of $x$ for the following expression:

$\left|\frac{x^2-3x+2}{x^2+3x+2}\right|\ge 1$

1. $(-\infty ,1)$
   

2. (0, 81)
   

3. $(-\infty ,-2)\cup (-2,-1)\cup (-1,0)$

4. None of these

Q. If the distance between -x and -y is 9 units, find the value of $|x-y|$. Where x and y are two points on the number line.
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Q.

Let S denote the set of real values of x for which
$|x^3-x|\le x\left(1\right)and\frac{2}{|x-2|}>\frac{3}{|1-2x|}\left(2\right)$
then S equals

1. $(2,\infty )$
   

2. $(\frac{8}{7},\sqrt{2})$
   

3. $(\sqrt{2},\infty )$
   

4. $(\frac{8}{7},\infty )$

Q.

If a>0 and the equation $a{x}^2+bx+c=0$ has two real roots $\alpha$ and $\beta$ such that $\left|\alpha \right|\le 1,\left|\beta \right|\le 1,$ then

1. $a+b+c\ge 0$
   

2. $a-b+c\ge 0$
   

3. $a+\left|b\right|+c\ge 0$
   

4. $a-c\ge 0$

Q.

Find the interval of $x$ such that $\left|\frac{x^2-3x-1}{x^2+x+1}\right|<3$ is satisfied is

1. $(-\infty ,-2)\cup (-1,\infty )$

2. None of these

3. $[\frac{8}{5},2]\cup [\frac{5}{2},\infty ]$

4. $(-5,-2)\cup (2,5)$