Final answer:
To solve the inequality, isolate the variable x by multiplying both sides of the inequality by the denominators. Then, simplify and solve for x by factoring. Test different regions of the number line to determine the intervals that satisfy the inequality.
Step-by-step explanation:
To solve the inequality, we need to isolate the variable x. Start by multiplying both sides of the inequality by (2x-3)(x+4) to get rid of the denominators. This gives us 6x(x+4) < 3x-1(2x-3). Simplify both sides of the equation and combine like terms. Then, solve for x by moving all the terms to one side and factoring.
Next, set up a number line and determine the critical values based on the factored inequality. Test different regions of the number line to determine which intervals satisfy the inequality. The solution to the inequality is the set of all x-values that make the inequality true.
For example, if we plug in a value less than the smaller critical value, the inequality will not hold. But if we plug in a value between the two critical values or greater than the larger critical value, the inequality will hold true. Therefore, the solution to the inequality is x > -4 and x < 3/2.