MathLabs

Problem 2

For which real numbers xx does the inequality 4x2(1−1+2x)2<2x+9\dfrac{4x^2}{\left(1-\sqrt{1+2x}\right)^2} < 2x+9 hold?
Step 4 of 5: Cancel the common factor and simplify
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(a+1)2<a2+8  ⟺  a<72(a+1)^2 < a^2+8 \iff a<\dfrac72
Detailed analysis

Since (a2−1)2=(a−1)2(a+1)2(a^2-1)^2=(a-1)^2(a+1)^2 and the denominator is (1−a)2=(a−1)2(1-a)^2=(a-1)^2, and a≠1a\neq1 so (a−1)2≠0(a-1)^2\neq0, dividing top and bottom by (a−1)2(a-1)^2 leaves (a+1)2<a2+8(a+1)^2<a^2+8. Expanding, a2+2a+1<a2+8a^2+2a+1<a^2+8, i.e. 2a<72a<7, i.e. a<72a<\tfrac72.