Question 80790
Given:
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{{{4x + y >= 4 }}}
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Begin by treating this as just an ordinary equation:
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{{{4x + y = 4}}}
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You can easily find a couple of points on this line.  Do this by first setting x equal to
zero. If you do that, all that is left is {{{y = 4}}}.  Now you know that the point (0,4) 
is on the graph. (This point is on the y-axis at +4). Next set y equal to zero and you get:
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{{{4x = 4}}}. Solve for x by dividing both sides by 4 and the result is:
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{{{x = 4/4 = 1}}}
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You now know that the point (1,0) is also on the line. (This point is on the x-axis
at +1.
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Since you have two points plotted on the coordinate system, you can use a straight
edge to extend a line through these two points.
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This is the graph of the line {{{4x + y = 4}}}. It should look like this:
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{{{graph(300, 300, -10, 10, -10, 10, -4x + 4)}}}
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Now let's return to the original inequality:
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{{{4x + y >= 4}}}
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Let's get this into the slope intercept form. Recall that the slope intercept form is 
{{{y >= mx + b}}} where b is the value of y at the point where the graph crosses the y-axis.
And m is the slope of the line.  We can get the inequality into this form by subtracting
4x from both sides.  When you do that the equation becomes:
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{{{y >= -4x + b}}}
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but recall from above that we found +4 was the value on the y-axis where the graph 
intersected. Substitute this value for b to get:
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{{{y >= -4x + 4}}}
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This form tells you that the line that is the graph has a slope of -4 and crosses the
y-axis at +4.  Note that this slope intercept equation tells you that for each point on
the line, y must have a value equal to or greater than the value of y associated with that
point on the line.
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Therefore, you can shade in the area of the graph above that line and also on that line.
Y can have values anywhere in that shaded area. And all the (x, y) points in that shaded
area (including the line) will satisfy the original inequality.
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Let's try a point in the shaded area, and see if it works.  Pick a point on the y-axis 
above +4 ... say the point (0, 6). That point is also on the y-axis and has the values
x = 0, y = 6.  It is in the shaded area above the line. Return to the original inequality
and substitute 0 for x and 6 for y.
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{{{4x + y >= 4 }}}
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Substituting 0 for x and 6 for y results in:
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{{{4*0 + 6 >= 4}}}
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This simplifies to:
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{{{6 >= 4}}}
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This is obviously true ... 6 is greater than 4. And since the point (0, 6) is in the shaded
area, it helps to convince us that we have identified the shaded region correctly.
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You can try points above the line, on the line, and below the line to check out the solution
we got.  You will see that points that are on the line and above the line make the original
inequality true. But points below the line will not make the inequality correct.
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Hope this helps you to understand the solution to inequalities of this type, how to find
them and how to represent them by shading on the entire graph.
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