Method of Intervals

Q. A natural number is chosen at random from the first 100 natural numbers. Then the probability, for the in-equation $x+\frac{100}{x}>50$ satisfied, is

1. $\frac{1}{20}$
2. $\frac{1}{3}$
3. $\frac{1}{4}$
4. $\frac{11}{20}$

Q.

The range of values of $x$ for which the inequality $\frac{3x-2}{5x-3}\ge 4$ is satisfied is given by

1. $(\frac{2}{3},\frac{3}{5})$

2. $[\frac{-8}{3},\frac{12}{5})$

3. $[\frac{10}{17},\frac{3}{5})$

4. $[\frac{-3}{5},\frac{12}{5}]$

Q.

What are the conditions under which the logarithm ${\log }_{2x}\left(x^2-1\right)$ is defined ?

1. x > 0

2. x > 1

3. x > 1 and x < -1

4. x < -1

Q.

$\text{Solve for x}:(x-2{)}^3(x-3\left)\right(x-5{)}^2<0$

1. (-∞, 2)∪(3, ∞)

2. (2, 3)

3. (3, ∞)

4. (5, ∞)

Q.

Find the set of solution in ${\log }_{\frac{1}{2}}\left(x^2-6x+12\right)\ge -2$

1. [2, 4]

2. ($-\infty$, 2]

3. [4, $\infty$)

4. [2, $\infty$)

Q. A natural number is chosen at random from the first 100 natural numbers. Then the probability, for the in-equation $x+\frac{100}{x}>50$ satisfied, is

1. $\frac{1}{20}$
2. $\frac{11}{20}$
3. $\frac{1}{3}$
4. $\frac{1}{4}$

Q.

Solve for $x$:

$3\left(9^x\right)<8\left(3^x\right)+3$

1. (0, 81)

2. None of these.

3. $(-\infty ,1)$

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

Q. The maximum possible value of $f\left(x\right)=\frac{1-x^2}{x^2+4}$ is

1. $\frac{1}{4}$
2. $\frac{1}{3}$
3. $\frac{1}{2}$
4. \N

Q. $\mathrm{The}\mathrm{set}\mathrm{of}\mathrm{possible}\mathrm{values}\mathrm{of}x\mathrm{for}\mathrm{which}\frac{2x}{\left(}\ge \frac{1}{x+1}\mathrm{is}/\mathrm{are}$

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

Q. The minimum value of x satisfying the inequality $4^{-x+0.5}-7\cdot 2^{-x}\le 4$ is