To solve the equation [tex]\( x + x \sqrt{2} = 10 \)[/tex] for [tex]\( x \)[/tex] and then round it to the nearest tenth, follow these steps:
1. Simplify the equation:
Start by factoring [tex]\( x \)[/tex] out of the equation:
[tex]\[
x (1 + \sqrt{2}) = 10
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
Divide both sides of the equation by [tex]\( (1 + \sqrt{2}) \)[/tex] to isolate [tex]\( x \)[/tex]:
[tex]\[
x = \frac{10}{1 + \sqrt{2}}
\][/tex]
3. Calculate the value of [tex]\( x \)[/tex]:
Compute the right-hand side value:
[tex]\[
x \approx 4.142135623730951
\][/tex]
4. Round the value to the nearest tenth:
Look at the digit in the hundredths place (4.142...). The hundredths place is 4, so the tenths place remains unchanged when rounding:
[tex]\[
x \approx 4.1
\][/tex]
Thus, the value of [tex]\( x \)[/tex] rounded to the nearest tenth is [tex]\( \boxed{4.1} \)[/tex].
