Question 1208860
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(2x-1)^(-1) is the same as 1/(2x-1)


As mentioned by the other tutor, 1/(2x-1) is positive if and only if 2x-1 > 0.
That solves to x > 1/2


A visual way to demonstrate this is to use a graphing tool like <a href="https://www.geogebra.org/calculator">GeoGebra</a> or <a href="https://www.desmos.com/calculator">Desmos</a> to plot the graph of  y = 1/(2x-1)
{{{
drawing(500,500,-5,5,-5,5,
graph(500,500,-5,5,-5,5,-100,1/(2x-1)),
line(0.5,-6,0.5,-5.4),line(0.5,-4.8,0.5,-4.2),line(0.5,-3.6,0.5,-3),line(0.5,-2.4,0.5,-1.8),line(0.5,-1.2,0.5,-0.6),line(0.5,0,0.5,0.6),line(0.5,1.2,0.5,1.8),line(0.5,2.4,0.5,3),line(0.5,3.6,0.5,4.2),line(0.5,4.8,0.5,5.4)
)
}}}
The vertical asymptote is x = 1/2 since this x value makes the denominator 2x-1 to be zero. Recall that dividing by zero is not allowed.


The portion of the curve to the right of x = 1/2 is entirely above the x axis. 
So we visually demonstrate the answer as an inequality is x > 1/2
You can verify this using a table of values. 


The answer in interval notation is (1/2, infinity)
Replace the word "infinity" with the symbol if needed.


The graph on a number line will involve an open hole at 1/2 and shading to the right. The open hole indicates "do not include this endpoint".
{{{
drawing(400,200,-6,6,-3,3,
line(-10,0,10,0),line(-5,0.04,-5,-0.04),line(-2.5,0.04,-2.5,-0.04),line(0,0.04,0,-0.04),line(2.5,0.04,2.5,-0.04),line(5,0.04,5,-0.04),locate(-5.1,-0.25,"-2"),locate(-2.6,-0.25,"-1"),locate(-0.05,-0.25,"0"),locate(2.45,-0.25,"1"),locate(4.95,-0.25,"2"),
red(
circle(5/4,0,0.18),circle(5/4,0,0.20),circle(5/4,0,0.22),circle(5/4,0,0.24),
line(5/4+0.2,0+0.08,6.123,0+0.08),
line(5/4+0.2,0+0.06,6.123,0+0.06),
line(5/4+0.2,0,6.123,0),
line(5/4+0.2,0-0.08,6.123,0-0.08),
line(5/4+0.2,0-0.06,6.123,0-0.06),
locate(5/4-0.5,0.2+0.5,"1/2"),
line(6+0,0,6-0.4+0,0-0.4),line(6+0,0,6-0.4+0,0+0.4),line(6+0.05,0,6-0.4+0.05,0-0.4),line(6+0.05,0,6-0.4+0.05,0+0.4),line(6+0.1,0,6-0.4+0.1,0-0.4),line(6+0.1,0,6-0.4+0.1,0+0.4),line(6+0.15,0,6-0.4+0.15,0-0.4),line(6+0.15,0,6-0.4+0.15,0+0.4),line(6+0.2,0,6-0.4+0.2,0-0.4),line(6+0.2,0,6-0.4+0.2,0+0.4)
)
)
}}}
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