Why x² < 2^x?

Steps to Solve

1. Rearranging the Inequality

We start with the inequality $x^2 < 2^x$. To analyze it, we can rearrange it to have one side equal to zero:

$$ x^2 - 2^x < 0 $$

2. Identifying Intersection Points

Next, we need to find points where the two sides of the inequality intersect, which will help us analyze the regions where one is less than the other. We can set them equal:

$$ x^2 = 2^x $$

3. Finding Roots Graphically or Numerically

To find the roots of the equation, we may graph it or test integer values:

For example:

• When $x = 0$: $0^2 = 0$ and $2^0 = 1$ → $0 < 1$
• When $x = 1$: $1^2 = 1$ and $2^1 = 2$ → $1 < 2$
• When $x = 2$: $2^2 = 4$ and $2^2 = 4$ → $4 = 4$
• When $x = 3$: $3^2 = 9$ and $2^3 = 8$ → $9 > 8$

It looks like the two functions intersect at $x = 2$.

4. Analyzing the Behavior of the Functions

To analyze the inequality, we can check intervals:

• For $x < 0$, for example $x = -1$: $$ (-1)^2 = 1 \text{ and } 2^{-1} = 0.5, \text{ so } 1 > 0.5 $$ (not satisfying)

• For $0 < x < 2$, we already checked a few values and found they satisfy the inequality:

• For $x > 2$, we already found from $x = 3$: $$ 3^2 = 9 \text{ and } 2^3 = 8, \text{ so } 9 > 8 $$ (not satisfying)

5. Conclusion on the Inequality Range

From our analysis, we find that the inequality $x^2 < 2^x$ holds true in the intervals:

$$(-\infty, 2)$$

Summary of Results:

• The inequality is satisfied for $x < 2$ and does not hold true for $x \geq 2$.