How do you graph the inequality #5/(x+3) ≥ 3/x#?

Question

Eva Adamson · Accepted Answer

The final graph will be the shape of #5/(x+3)# with #-3 <=x <=0; x>=4.5#

In order to graph the whole relationship, we need to graph both the left and right individually.

Here's the left side #(5/(x+3))#:

graph{5/(x+3) [-20, 20, -20, 20]}

And the right #(3/x)#:

graph{3/x [-20, 20, -20, 20]}

So now the question is, for what values of #x# is #5/(x+3)>=3/x#. We'll find that wherever the result of #5/(x+3)#, in essence the #y# value, is larger than that of #3/x#:

graph{(y-(5/(x+3)))(y-3/x)=0 [-20, 20, -20, 20]}

By observation, we can see that, approaching the graph from the left to the right, that the first value of #x# where this holds is at #x=-3# and continues until #x=0#.

At #x=4.5#, it holds again and does so until #+oo#:

graph{(y-(5/(x+3)))(y-3/x)=0 [4, 6, 0, 1]}

And so the final graph will be the shape of #5/(x+3)# with #-3 <=x <=0; x>=4.5#

Eva Abramson · Answer

undefined To graph the inequality 5/(x+3) ≥ 3/x, follow these steps:

1. Find the critical points by setting the expression equal to zero and solving for x.
2. Test the intervals determined by the critical points to see where the inequality holds true.
3. Plot the solutions on a number line to represent the graph of the inequality.

Critical points:
1. Set 5/(x+3) = 3/x and solve for x.
2. Find the values of x that make the denominator equal to zero, as they are not in the domain of the expression.

Test intervals:
1. Choose test points within each interval determined by the critical points.
2. Substitute these test points into the original inequality to see where it holds true.

Graph the solution:
1. Plot the critical points and test points on a number line.
2. Shade the intervals where the inequality holds true.

This process will give you the graph of the inequality 5/(x+3) ≥ 3/x.