Solving the Inequality (2/x - 4 > 3x 1/x - 3)

When dealing with algebraic inequalities, it is important to break down the problem step-by-step to understand the behavior of the variables involved. Consider the following inequality:

2/x - 4 > 3x 1/x - 3

Step 1: Simplify the Equation

Let us start by simplifying the given inequality. Suppose we have:

2/x - 4 3x 1/x - 3 …………….[M]

By rearranging the terms, we can get:

3x 1/x 2/x - 4 3

Simplifying further:

3x 1/x 2/x - 1

Multiplying both sides by x to eliminate the denominators:

3x2 1 2 - x

Transposing terms to one side:

3x2 x - 1 0

Step 2: Solving the Quadratic Equation

The equation can be solved using the quadratic formula, which states:

x [-b ± √(b2 - 4ac)] / (2a)

In this case, the coefficients are:

Substituting the values into the quadratic formula:

x [-1 ± √(1 12)] / 6 ≈ [-1 ± √13] / 6

This results in the solutions:

Step 3: Analyzing the Behavior of the Inequality

Now, to determine the range of x for which the inequality holds, we need to test values in the intervals around the critical points.

Testing for x 1 and x 0.434

When x 1:

2/1 - 4  3(1)   1/1 - 32 - 4  3   1 - 3-2  1

It does not hold true, hence -2 1. Therefore, LS RS.

Testing for x -1 and x -0.768

When x -1:

2/-1 - 4  3(-1)   1/-1 - 3-2 - 4  -3 - 1 - 3-6  -7

It also does not hold true, and -6 -7. Therefore, LS RS.

Conclusion

From our analysis, we can conclude that the inequality:

2/x - 4 > 3x 1/x - 3

holds true for values of x between the critical points:

-1 ± √13/6

Thus, the precise answer is:

2/x - 4 > 3x 1/x - 3 so long as x ∈ [-1 ± √13]/6.