Find the Domain

Problem

√(,√(,15*x+19))=√(,2*x+3)

Solution

1. Identify the conditions for the square root functions to be defined. For any expression √(,u) the radicand must be non-negative: u≥0

2. Set up the inequality for the innermost radicand on the left side:

15*x+19≥0

3. Solve for x in the first inequality:

15*x≥−19

x≥−19/15

4. Set up the inequality for the outer radicand on the left side. Since √(,15*x+19) is always non-negative whenever it is defined, the condition √(,15*x+19)≥0 is satisfied for all x≥−19/15

5. Set up the inequality for the radicand on the right side:

2*x+3≥0

6. Solve for x in the second inequality:

2*x≥−3

x≥−3/2

7. Determine the intersection of the two intervals. We compare the values −19/15≈−1.266 and −3/2=−1.5 Since −1.266>−1.5 the more restrictive condition is x≥−19/15

Final Answer

Domain:x≥−19/15

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