(2^x-1-1)/(2^x+1-1)<2

Name: Problem 291, Chapter 7: Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry
Uploaded: 2022-03-03T00:21:56-08:00
Duration: 3 min 38 s
Channel: Kavita Kumari
Description: (2^x-1-1)/(2^x+1-1)<2

Key Concepts

Domain Restrictions

When working with rational expressions, it is important to identify domain restrictions by finding values that make the denominator zero, because these values are not allowed. Ensuring that the expression is defined is crucial for correctly solving and interpreting the inequality.

Exponential Functions

Exponential functions are mathematical expressions in which the variable appears in the exponent. Their special properties, such as rapid growth or decay, make them distinct from linear or polynomial functions and require unique techniques for analysis and manipulation.

Properties of Exponents

The properties of exponents allow for the simplification and manipulation of expressions where variables appear as exponents. This includes rules such as product of powers, power of a power, and quotient of powers, which are key in re-writing and simplifying expressions in exponential equations and inequalities.

Rational Inequalities

Rational inequalities involve expressions that are ratios of two functions. Solving these inequalities typically requires careful analysis of both the numerator and the denominator, taking into account where the expressions change sign and identifying critical points that define different regions of the solution.

Inequality Manipulation

Manipulating inequalities, especially those involving rational or exponential expressions, requires caution. Particularly, when multiplying or dividing both sides of an inequality by an expression, one must consider the sign of that expression as it might reverse the inequality direction if it is negative.

Transcript

00:01 In the question, we have been given the inequality as 2 raised to the power x minus 1 minus 1 upon 2 raised to the power x plus 1 minus 1.

00:13 And this entire expression is less than 2.

00:17 We have to find out what can be the possible values of x which will satisfy this equation.

00:23 So firstly we will be applying the exponent's rule, the exponents rule and this rule is a very simple rule which you all might be aware of so this can be written as 2 ratio to the power x upon 2 minus 1 upon 2 raised to the power x into 2 minus 1 so this is how you can break this down and this entire thing is less than 2 now let us take a substitution here.

01:00 So let 2 raise to the power x be equal to u.

01:05 So the next step here will be, wherever we see 2 raise to the power x, we will substitute it with u.

01:11 So u by 2 minus 1 upon 2 u minus 1 is less than 2.

01:18 So this is a simple two step method that we've used here.

01:23 Now we'll be simplifying and find out what can be the possible value or ranges or intervals for you.

01:30 So on calculating, we can find out that this is a simple calculation.

01:36 So you here will be less than 2 by 7.

01:39 There are two possibilities.

01:41 And even we have to take into account the denominator or you will be greater than 1 by 2.

01:46 Only then this number will be less, this entire expression, the left hand side of this in equal.

01:53 Quality will be less than 2.

01:55 So these are the two values, two ranges of u...

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