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There are two critical points for the inequality to consider: x = – 5 and x = 3/2
Region I: x is greater than 3/2
In this scenario, both the terms would be positive; cross-multiplying, we get the relation 2x – 3 ≤ x + 5
Giving the boundary x ≤ 8, hence giving us the valid range as 3/2 < x ≤ 8
Region II: – 5 < x < 3/2
In this case, the right-hand side will be a negative value, and hence, the sign would change when multiplying, giving the inequality
2x – 3 ≥ x + 5
Which will give x > 8, which is out of bounds for this region
Another way is to put a value in the region to check for the validity of the inequality; by putting x = 0, we could see that the inequality does not hold in this region
Region III: x less than – 5
In this scenario, both the terms are negative, essentially giving us the same boundary as region 1; we take the lower bounds, giving us that x has to be less than 5
Therefore, for the given inequality to hold true x < – 5 or 3/2 < x ≤ 8