Exercise 81 Page 442

Before solving the radical inequality, recall that the radicand of a square root must be greater than or equal to 0. sqrt(3x+1) - sqrt(6+x) > 0 Let's start by identifying the values of x for which the square roots are defined. sqrt(3x+1):& 3x+1 ≥ 0 ⇔ x ≥ - 1/3 sqrt(6+x):& 6+x ≥ 0 ⇔ x ≥ - 6After solving the given inequality, we will combine the solution with the above intervals to obtain the final solution set. Let's rearrange the equation such that one variable term is on either side of the inequality. ccc sqrt(3x+1) - sqrt(6+x) > 0 [0.5em] ⇕ [0.5em] sqrt(3x+1)>sqrt(6+x) Our next step would be to square both sides of the above inequality. To do this, we have to be sure both sides are non-negative. Otherwise, we may get incorrect solutions.

In this case, both sides will always be non-negative so we can proceed with squaring both sides of the inequality.

Now we can combine the three intervals.

The intersection of the three solutions, which is the final solution set of the given inequality, is x > 52.