Please simplify and show the complete solution. Thank you so much! To be clear the first term is 5 over 7a - 3 plus 3 over a less than 5a? To be exact its in rational form.
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1 Answer

(5/7a)-3+3/a<5a. Multiply through by 7a: 5-21a+21<35a^2 assuming a>0.

35a^2+21a-26>0 has irrational roots one of which is negative, which reverses the inequality.

The positive root is 0.61261 approx, and the negative root is -1.21261.

To satisfy the inequality, a<-1.21261 or a>0.61261.

To appreciate this better, 35a^2+21a-26 is a parabola which cuts the a axis (horizontal) at the roots. It's U-shaped so the vertex is below the intersection points. This means the parabola equation is negative between the roots, so it's positive outside the roots. Therefore a<negative root and a>positive root satisfies the inequality.

It's possible the question is supposed to read differently. Parentheses help to make the intention clear. If the first term is 5/(7a-3) the result is a cubic which doesn't factorise and the solution is more complicated involving an irrational root.

Another interpretation: 5/(7a-3+(3/a))<5a; 5a/(7a^2-3a+3)<5a; 1/(7a^2-3a+3)<1; 1<7a^2-3a+3; 7a^2-3a+2>0. This has no real roots and is always positive for all a. So this interpretation is going nowhere!

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