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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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