Find the Domain 9x^4+10y^4=42

Problem

9*x^4+10*y^4=42

Solution

1. Isolate the term containing y to express y as a function of x

10*y^4=42−9*x^4

2. Divide both sides by 10 to solve for y^4

y^4=(42−9*x^4)/10

3. Apply the fourth root to both sides. Since the index is even, the expression inside the root must be greater than or equal to zero for y to be a real number.

(42−9*x^4)/10≥0

4. Solve the inequality for x by multiplying by 10 and rearranging the terms.

42−9*x^4≥0

9*x^4≤42

5. Simplify the inequality by dividing by 9

x^4≤42/9

x^4≤14/3

6. Take the fourth root of both sides to find the interval for x

|x|≤√(4,14/3)

−√(4,14/3)≤x≤√(4,14/3)

Final Answer

Domain:[−√(4,14/3),√(4,14/3)]

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