How do you graph the inequality #2(y-1) > 3(x+1)#?

Answer 1

See below:

One way to do it is to put the inequality into a form that is more recognizable as the graph of a line. For instance, I prefer to use the slope intercept form, so let's put this inequality into that form:

#2(y-1)>3(x+1)#
#2y-2>3x+3#
#2y>3x+5#
#y>3/2x+5/2#
If we ignore the inequality sign for a moment and graph the line #y=3/2x+5/2#, we get this:

graph{3/2x+5/2}

So now the question is to decide which side of the line we need to shade. We can do that by seeing whether the point #(0,0)# satisfies the inequality (we can use any point, but the origin is often quite easy to use):
#0>3/2(0)+5/2=>0>5/2color(white)(000)color(red)X#

And so we shade the side of the line that does not have the origin in it.

The line itself will be dotted to indicate that the points on the line do not satisfy the inequality:

graph{y-3/2x-5/2>0}

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Answer 2

To graph the inequality (2(y-1) > 3(x+1)), we can start by rewriting it in slope-intercept form ((y = mx + b)):

[ 2(y-1) > 3(x+1) \ 2y - 2 > 3x + 3 \ 2y > 3x + 5 \ y > \frac{3}{2}x + \frac{5}{2} ]

Now, we can graph the line (y = \frac{3}{2}x + \frac{5}{2}) using its slope ((m = \frac{3}{2})) and y-intercept ((b = \frac{5}{2})).

  1. Plot the y-intercept at ((0, \frac{5}{2})).
  2. Use the slope to find another point. Since the slope is (\frac{3}{2}), we can move up 3 units and right 2 units from the y-intercept to get another point.
  3. Draw a dashed line through these two points.

Since the inequality is (y > \frac{3}{2}x + \frac{5}{2}), the region above the line is shaded.

So, the graph of the inequality (2(y-1) > 3(x+1)) is a dashed line with the area above the line shaded.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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