Problem 2

For which real numbers

x

does the inequality

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

hold?

Step 1 of 5: Domain restrictions from the square root and the denominator

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In plain words

Before touching the inequality itself, pin down exactly which $x$ even make sense to plug in — a square root and a denominator each forbid part of the real line.

1+2x\ge 0 \text{ and } \sqrt{1+2x}\neq 1

Detailed analysis

The square root $\sqrt{1+2x}$ is only real when $1+2x\ge0$ , i.e. $x\ge-\tfrac12$ . The denominator $\left(1-\sqrt{1+2x}\right)^2$ must also be nonzero, i.e. $\sqrt{1+2x}\neq1$ , i.e. $1+2x\neq1$ , i.e. $x\neq0$ . These are exactly the values of $x$ for which the left-hand side is even defined.