Problem 2

For which real numbers

x

does the inequality

\dfrac{4x^2}{\left(1-\sqrt{1+2x}\right)^2} < 2x+9

hold?

Step 5 of 5: Translate back to x and state the final answer

Previous step Next step

-\dfrac12\le x<\dfrac{45}{8},\quad x\neq0

Detailed analysis

The map $a\mapsto x=\tfrac{a^2-1}{2}$ is strictly increasing for $a\ge0$ , sending $a=0\mapsto x=-\tfrac12$ , $a\to\tfrac72\mapsto x\to\tfrac{45}{8}$ , and $a=1\mapsto x=0$ . Combining $0\le a<\tfrac72$ with $a\neq1$ therefore gives exactly $-\tfrac12\le x<\tfrac{45}{8}$ with $x\neq0$ .