Final answer:
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.