Question 352194
Hi,
*Note: In this case one cannot multiply the right hand side by (2x-3) because the value of x is unknown. Since x may be either positive or negative, you can't know whether to leave the inequality sign as <, or reverse it to >. 
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When Solving an inequality with a variable in the denominator:
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First Step:
Find when the denominator would equal zero:
2x + 3 = 0
x = -3/2
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Second step:
Examine the results of replacing > with an = sign. That results in a false statement 0 = 6 as you found. As there is no solution when that = sign is used:

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x < -3/2 are the only numbers that work in this inequality