Question
\( 1 \leftarrow \quad \begin{array}{l}\text { Solve. } \\ \sqrt{9 x-2}-5=0\end{array} \)
Ask by Ray Greene. in the United States
Mar 14,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x=3 \)
Solution
Solve the equation \( \sqrt{9x-2}-5=0 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(\sqrt{9x-2}-5=0\)
- step1: Find the domain:
\(\sqrt{9x-2}-5=0,x\geq \frac{2}{9}\)
- step2: Move the constant to the right side:
\(\sqrt{9x-2}=0+5\)
- step3: Remove 0:
\(\sqrt{9x-2}=5\)
- step4: Raise both sides to the \(2\)-th power\(:\)
\(\left(\sqrt{9x-2}\right)^{2}=5^{2}\)
- step5: Evaluate the power:
\(9x-2=25\)
- step6: Move the constant to the right side:
\(9x=25+2\)
- step7: Add the numbers:
\(9x=27\)
- step8: Divide both sides:
\(\frac{9x}{9}=\frac{27}{9}\)
- step9: Divide the numbers:
\(x=3\)
- step10: Check if the solution is in the defined range:
\(x=3,x\geq \frac{2}{9}\)
- step11: Find the intersection:
\(x=3\)
- step12: Check the solution:
\(x=3\)
The solution to the equation \( \sqrt{9x-2}-5=0 \) is \( x=3 \).
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Simplify this solution Beyond the Answer
To solve the equation \( \sqrt{9x - 2} - 5 = 0 \), we can start by isolating the square root. This gives us \( \sqrt{9x - 2} = 5 \). Next, we square both sides to eliminate the square root, resulting in \( 9x - 2 = 25 \). Now, add 2 to both sides: \( 9x = 27 \). Finally, divide by 9 to find \( x = 3 \). Remember to check your solution! Substituting \( x = 3 \) back into the original equation, \( \sqrt{9(3) - 2} - 5 = \sqrt{27 - 2} - 5 = \sqrt{25} - 5 = 5 - 5 = 0 \), confirms that our solution works perfectly!
