Graph of linear inequalities in two variables

Question

Graph the inequality y≥9/2x+9

 

I am trying to figure out where 9/2 would go on a graph. 

Answer 1

Hi Shawna,

 

Where is the x in your inequality?

 

If the x is in the fraction on bottom, then follow Russ P's advice. 

 

If the x is in the fraction on top, then follow Francisco P's advice.

 

If the x is not in the fraction but just to the side of the fraction, then follow Francisco P's advice.

 

Hope this helps!

Answer 2

Shawna,

I assume that the inequality throws you for a loop and I further assume your inequality is really: y>= (9/(2x)) + 9 to be utterly explicitly by using parentheses to control the precedence of operations.

 

The best approach is to two step this problem.  First graph the equality curves since they are part of and boundaries for the solution.  Then apply the inequality to see which "side" of the curve you're on.

 

As x gets large positive or negative, the term 9/(2x) approaches zero, so the curves must be asymptotic to y = 9.

As x approaches zero, then the term 9/(2x) blows up either positive for positive x or negative for negative x.

So the graph has two parts to it. As x starts out at very large negative, y is below y=9 and asymptotic to it, then bends down as negative x approaches zero, becoming asymptotic to the negative y axis.

The other part, starts out above y=9 & asymptotic for very large positive x, then as x approaches zero it bends upward, and becomes asymptotic to the positive y axis as y goes to plus infinity.

 

Now apply the inequality part. For positive x, the inequality is true whenever y is above the curve.  For example at x=1, y=4.5 + 9 =13.5 on the curve. But any y>13.5 satisfies the inequality condition at x=1.

For negative x, all the y values (positive and negative) wedged between the curve and the y=9 asymptote satisfy the inequality.  For example, at x=-1. y= -4.5 + 9 = 4.5 so its below the y=9 line and above the curve.  They can't be above the y=9 line because the 9/(2x) term is always negative for negative x.  And they can't be below the curve because the +9 term always raises the total above the line.

 

In summary, when somethings appears complex, simplify it by breaking it down into smaller, more familiar and manageable steps.  Einstein said "complexity is nothing more than structured simplicity".  Good luck.

Answer 3

9/2 is the slope which would help you plot the line.

 

Go 2 units right, then 9 units up starting at (0,9)--because of the value of the intercept--to get to (2,18).  Or you could go 2 units left and 9 units down from (0,9) to get to (-2,0).

 

Once you get two or more points, draw a straight line between them. 

 

You can test a few points in the x-y plane to see if they satisfy the inequality.  For example, the origin would make the statement false, so you would shade in the side of the x-y plane divided by the line where the origin is not located.