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## Exponential Inequalities

When will a bacteria colony reach a certain size? When will one investment outperform another? Many real-world models use exponential functions, and you'll need these tools to compare them.

# Problem Solving

How many positive integers $$x<1000$$ satisfy the inequality $$x^{3x+1} > x^{x+139}$$?

Which of the following satisfies the inequality $3^y \geq 4^x - 2 ?$

How many integers $$x$$ satisfy the inequality $\frac{2 ^5}{5 ^5} < \left( \frac{2}{5} \right)^{x^2} < \frac{125}{8} \cdot \left( \frac{16}{625} \right)^x ?$

What is the minimum value of $$x$$ that satisfies the inequality $$2^{2x} - 127 \cdot 2^x -128 \geq 0$$?

If $$y=f(x)$$ is the equation of the line which has a slope of $$-\frac{1}{5}$$ and passes through the point $$(0,5),$$ what is the solution to the inequality $0 \leq f(5^x) \leq 10 ?$

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