Question

What is the solution set for the exponential inequality 1/(2²ˣ⁺¹) < 4?

Answer 1

The solution set for the exponential inequality 1/(2²⁴⁺¹) < 4 is x > -3/2, which represents all numbers greater than -1.5.

Step-by-step explanation:

To solve the exponential inequality 1/(2²⁴⁺¹) < 4, we first need to rewrite the inequality in a more manageable form. To do this, we can express the number 4 as a power of 2 since this will allow us to compare the exponents directly. We know that 4 is equal to 2², and using this we can rewrite our inequality as:

1/(2²⁴⁺¹) < 2²

Now, we take the reciprocal of both sides, bearing in mind that this will reverse the inequality sign:

2²⁴⁺¹ > 1/2²

This simplifies to:

2²⁴⁺¹ > 2⁻²

Now we have the same base on both sides, which allows us to compare the exponents directly. This gives us the inequality:

2x+1 > -2

Subtracting 1 from both sides, we obtain:

2x > -3

Dividing both sides by 2, we arrive at the solution for the inequality:

x > -3/2

So the solution set for the inequality is x > -3/2, meaning that any number greater than -1.5 will satisfy the original exponential inequality.