Inequalities involving modulus function

Q.

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

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

1. (0, 81)
2. $(-\infty ,1)$

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

4. None of these

Q. The sum of all real roots of the equation ${\left(x-2\right|}^2+\left(x-2\right|-2=0$ is

Q.

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

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

2. -13 $\le$ x $\le$ 17

3. -5 $\le$ x $\le$ 27

4. -5 $\le$ x $\le$ 17

Q. Which of the following is/are the solution(s) of the inequality $\frac{1}{\left(x\right|-3}\ge \frac{1}{2}$?

1. $\left(-\infty ,-5\right)$
2. $[-5,-3)$
3. $\left(-3,3\right)$
4. $(3,5]$
5. $\left(5,\infty \right)$

Q.

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

1. 5

2. 2

3. 8

4. 1

Q. The number of integral solutions of the linear inequality 1 ≤ |x –2 | ≤ 3 is

1. 4
2. 5
3. 6
4. 7
5. 9
6. 8

Q. If x satisfies $\left(x-1\right|+\left(x-2\right|+\left(x-3\right|\ge 6$, then

1. $0\le x\le 4$
2. $x\le -2\mathrm{or}x\ge 4$
3. $x\le 0\mathrm{or}x\ge 4$
4. $-2\le x\le 4$

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

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

4. None of these

Q.

Solution set of $lo{g}_{0.25}(x^2+2x-8{)}^2-lo{g}_{0.5}(10+3x+x^2)=1$

1. $\left\{\frac{1}{4}\right\}$
2. $\{-3\}$
3. $\left\{\frac{1}{6}\right(\sqrt{313}-1),\frac{1}{2}(\sqrt{73}-7\left)\right\}$
4. $\{1,7\}$

Q. If $|x-1|\le 2$
then, the value of x is more than equal to and less than equal to .

1. -2
2. -1
3. -3
4. 1
5. 2
6. 3

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. $a-c\ge 0$