Question:
The set of real values of x for which $\log_{0.2} \frac{x +2}{x} \leq 1$ is
The set of real values of x for which $\log_{0.2} \frac{x +2}{x} \leq 1$ is
Updated On: Jul 7, 2022
- $\bigg( - \infty , - \frac{5}{2} \bigg] \cup (0, \infty)$
- $\bigg[ \frac{5}{2} , \infty \bigg) $
- $( - \infty , -2) \cup [0 , \infty)$
- none of these
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The Correct Option is A
Solution and Explanation
The inequality is $ {log_{0.2} \frac{x + 2}{x} \leq 1}$
The L.H.S is valid if $ { \frac{x + 2}{x} > 0}$
$ \Rightarrow \: {x(x + 2) > 0 \Rightarrow x < -2 \, or \: x > 0}$
Solving the inequality, we get (note that base < 1).
$ { \frac{x + 2}{x}\geq 0.2 = \frac{1}{5}}$
$\Rightarrow \frac{x +2}{x} -\frac{1}{5} \ge 0 \Rightarrow \frac{4x + 10}{5x} \ge 0$
$x \left(2x+5\right)\ge0 \Rightarrow x \le -\frac{5}{2}$ or $x \ge0$
Taking the intersection, we get
$ {x \leq - \frac{5}{2}}$ or $x > 0$
$\Rightarrow \: x \in \left( - \infty , - \frac{5}{2} \right] \cup (0 , \infty)$
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Concepts Used:
Sets
Set is the collection of well defined objects. Sets are represented by capital letters, eg. A={}. Sets are composed of elements which could be numbers, letters, shapes, etc.
Example of set: Set of vowels A={a,e,i,o,u}
Representation of Sets
There are three basic notation or representation of sets are as follows:
Statement Form: The statement representation describes a statement to show what are the elements of a set.
- For example, Set A is the list of the first five odd numbers.
Roster Form: The form in which elements are listed in set. Elements in the set is seperatrd by comma and enclosed within the curly braces.
- For example represent the set of vowels in roster form.
A={a,e,i,o,u}
Set Builder Form:
- The set builder representation has a certain rule or a statement that specifically describes the common feature of all the elements of a set.
- The set builder form uses a vertical bar in its representation, with a text describing the character of the elements of the set.
- For example, A = { k | k is an even number, k ≤ 20}. The statement says, all the elements of set A are even numbers that are less than or equal to 20.
- Sometimes a ":" is used in the place of the "|".