Question
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Tutor-Verified Answer
Answer
\( x=1 \)
Solution
Solve the equation \( \sqrt{9x+7}-4=0 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(\sqrt{9x+7}-4=0\)
- step1: Find the domain:
\(\sqrt{9x+7}-4=0,x\geq -\frac{7}{9}\)
- step2: Move the constant to the right side:
\(\sqrt{9x+7}=0+4\)
- step3: Remove 0:
\(\sqrt{9x+7}=4\)
- step4: Raise both sides to the \(2\)-th power\(:\)
\(\left(\sqrt{9x+7}\right)^{2}=4^{2}\)
- step5: Evaluate the power:
\(9x+7=16\)
- step6: Move the constant to the right side:
\(9x=16-7\)
- step7: Subtract the numbers:
\(9x=9\)
- step8: Divide both sides:
\(\frac{9x}{9}=\frac{9}{9}\)
- step9: Divide the numbers:
\(x=1\)
- step10: Check if the solution is in the defined range:
\(x=1,x\geq -\frac{7}{9}\)
- step11: Find the intersection:
\(x=1\)
- step12: Check the solution:
\(x=1\)
The solution to the equation \( \sqrt{9x+7}-4=0 \) is \( x=1 \).
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Simplify this solution Beyond the Answer
Let’s simplify that equation! First, rearrange it to isolate the square root: \( \sqrt{9x + 7} = 4 \). Squaring both sides gives us \( 9x + 7 = 16 \). Now subtract 7 from both sides to get \( 9x = 9 \), and divide by 9 to find \( x = 1 \). Voila! Keep in mind that when you deal with square roots, it's always good practice to check your solution by substituting it back into the original equation. Plugging in \( x = 1 \) gives \( \sqrt{9(1) + 7 - 4} = \sqrt{12} \), which indeed equals 4 when simplified properly. Always double-check your steps!
