Question 1206316
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We can use a spreadsheet to check each answer mentioned (x = -pi/2 and x = -pi/4)
<table border = "1" cellpadding = "5"><tr><td>x</td><td>LHS</td><td>RHS</td><td>LHS = RHS?</td></tr><tr><td>-pi/2</td><td>0.414214</td><td>1</td><td>No</td></tr><tr><td>-pi/4</td><td>0.707107</td><td>0.707107</td><td>Yes</td></tr></table>
The table shows that only x = -pi/4 works.
LHS = left hand side
RHS = right hand side


Furthermore,
-pi/2 < -pi/4 < pi/2
Multiply all sides by -4
2pi > pi > -2pi
which flips to
-2pi < pi < 2pi


The last inequality 
-2pi < pi < 2pi
being true leads back to 
-pi/2 < -pi/4 < pi/2
also being true


Or you could use these decimal approximations
pi/2 = 1.570796
pi/4 = 0.785398
So
-pi/2 < -pi/4 < pi/2
becomes
-1.570796 < -0.785398 < 1.570796


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Another way to verify the answer is to graph h(x) = f(x) - g(x)
where
f(x) = sin(x) + sqrt(2)
g(x) = -sin(x)


Or basically you need to graph h(x) = 2*sin(x)+sqrt(2)
<a href="https://www.desmos.com/calculator/s6agjxpf7d">https://www.desmos.com/calculator/s6agjxpf7d</a>
Desmos is free graphing software. GeoGebra is also another good choice.
The curve intersects the x axis at (-pi/4, 0) to confirm that x = -pi/4 is the only solution to this equation on this specified domain interval.


If the -pi/2 < x < pi/2 portion wasn't required, then there would be infinitely many solutions.
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