Solving Inequalities Involving Square Roots: A Comprehensive Guide

When faced with inequalities that involve square roots, such as sqrt{x-1x3} sqrt{x-2x-3}, it is essential to follow a systematic approach to ensure accurate solutions. This guide will walk you through the steps to solve this specific inequality, including identifying domains, squaring both sides, and simplifying the inequality without altering its solution set.

Step 1: Identifying the Domains

To begin with, we need to identify the domains for which both sides of the inequality are defined and non-negative. The domain is crucial because it ensures that squaring the expressions is a valid operation.

Domain of the Left Side

The left side of the inequality is sqrt{x-1x3}. This expression is defined when x-1x3 > 0, which can be rewritten as x-1 and x3. The solutions to this inequality are:

- When x-1 > 0, we get x > 1.

- When x3 > 0, we get x .

Therefore, the domain for the left side is:

x x > 1

Domain of the Right Side

The right side of the inequality is sqrt{x-2x-3}. This expression is defined when x-2x-3 > 0, which can be rewritten as x-2 and x3. The solutions to this inequality are:

- When x-2 > 0, we get x > 2.

- When x3 > 0, we get x .

Therefore, the domain for the right side is:

x x > 3

Step 2: Combining the Conditions

The valid regions for both sides to be defined and non-negative are the intersections of the domains we found:

x 1 x > 3

Combining these valid regions, we get three intervals:

x 1 x > 3

Step 3: Squaring Both Sides of the Inequality

Now that we have identified the domains, we can proceed to square both sides of the inequality. This step ensures that we do not alter the solution set, provided both sides are non-negative within the valid regions. Therefore, we have:

x-1x3 - x-2x-3

Squaring both sides:

x^2 3x - x - 3 - x^2 - 3x - 2x 6.

Expanding and simplifying:

- Right Side: x^2 - 3x - 2x 6 x^2 - 5x 6.

- Left Side: x^2 3x - x - 3 x^2 2x - 3.

The inequality becomes:

x^2 2x - 3 - x^2 5x - 6.

Further simplification:

2x - 3 - 5x - 6 -3x - 9.

Thus, the simplified inequality is:

x^2 2x - 3 - x^2 5x - 6 -3x - 9.

Step 4: Simplifying the Inequality

By combining like terms:

2x - 3 - 5x - 6 -3x - 9.

Which simplifies to:

-3x - 9 0.

Finally, we divide both sides by -3:

x 9/7.

Since we need to consider the intervals we identified earlier, we must ensure that

x 9/7

satisfies the conditions for the square roots being defined:

For the left side, x > 1. For the right side, x > 3.

Given that 9/7 approx; 1.29, it does not satisfy the condition for the right side, which requires x > 3. However, it does satisfy the condition for the left side, x > 1.

Step 5: Final Solution

Therefore, the solution to the inequality is:

9/7 3.

In boxed form, the final solution set is:

boxed{9/7 3}

Conclusion

This comprehensive guide on solving inequalities involving square roots should provide you with a systematic approach to handle similar problems. Remember that identifying the domains and ensuring the expression is non-negative are crucial steps. Following these steps will help you accurately solve these types of inequalities.