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Sagot :
Let's solve the equation [tex]\(\sqrt{x-5} + 7 = 11\)[/tex] step by step.
1. Isolate the square root term:
[tex]\[ \sqrt{x-5} + 7 = 11 \][/tex]
Subtract 7 from both sides of the equation to isolate the square root term:
[tex]\[ \sqrt{x-5} = 11 - 7 \][/tex]
Simplify the right-hand side:
[tex]\[ \sqrt{x-5} = 4 \][/tex]
2. Eliminate the square root:
Square both sides of the equation to eliminate the square root:
[tex]\[ (\sqrt{x-5})^2 = 4^2 \][/tex]
Simplify:
[tex]\[ x-5 = 16 \][/tex]
3. Solve for [tex]\(x\)[/tex]:
Add 5 to both sides:
[tex]\[ x = 16 + 5 \][/tex]
Simplify:
[tex]\[ x = 21 \][/tex]
4. Verify the solution:
To ensure that [tex]\(x = 21\)[/tex] is not an extraneous solution, substitute [tex]\(x = 21\)[/tex] back into the original equation:
[tex]\[ \sqrt{21-5} + 7 = 11 \][/tex]
Simplify inside the square root:
[tex]\[ \sqrt{16} + 7 = 11 \][/tex]
Since [tex]\(\sqrt{16} = 4\)[/tex], we get:
[tex]\[ 4 + 7 = 11 \][/tex]
This simplifies to:
[tex]\[ 11 = 11 \][/tex]
The original equation holds true, so the solution [tex]\(x = 21\)[/tex] is correct and it is not extraneous.
Thus, the correct answer is:
[tex]\[ x = 21, \text{ solution is not extraneous } \][/tex]
1. Isolate the square root term:
[tex]\[ \sqrt{x-5} + 7 = 11 \][/tex]
Subtract 7 from both sides of the equation to isolate the square root term:
[tex]\[ \sqrt{x-5} = 11 - 7 \][/tex]
Simplify the right-hand side:
[tex]\[ \sqrt{x-5} = 4 \][/tex]
2. Eliminate the square root:
Square both sides of the equation to eliminate the square root:
[tex]\[ (\sqrt{x-5})^2 = 4^2 \][/tex]
Simplify:
[tex]\[ x-5 = 16 \][/tex]
3. Solve for [tex]\(x\)[/tex]:
Add 5 to both sides:
[tex]\[ x = 16 + 5 \][/tex]
Simplify:
[tex]\[ x = 21 \][/tex]
4. Verify the solution:
To ensure that [tex]\(x = 21\)[/tex] is not an extraneous solution, substitute [tex]\(x = 21\)[/tex] back into the original equation:
[tex]\[ \sqrt{21-5} + 7 = 11 \][/tex]
Simplify inside the square root:
[tex]\[ \sqrt{16} + 7 = 11 \][/tex]
Since [tex]\(\sqrt{16} = 4\)[/tex], we get:
[tex]\[ 4 + 7 = 11 \][/tex]
This simplifies to:
[tex]\[ 11 = 11 \][/tex]
The original equation holds true, so the solution [tex]\(x = 21\)[/tex] is correct and it is not extraneous.
Thus, the correct answer is:
[tex]\[ x = 21, \text{ solution is not extraneous } \][/tex]
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