# How do you solve #(x-1)/(2x+3) <=(1)#?

##### 2 Answers

The solution set is:

#### Explanation:

Start by comparing some function (expression) to zero.

[Note: this is not really necessary, but is simplifies the question to be asked later when is the expression of the left less that the number on the right. If the number on the right is

Combine the expression to get a single polynomial or rational expression. (In this question, it will be a rational expression.)

Now, if the expression on the left is going to change from positive (

The number that make the left side

(These numbers are sometimes called "Partition Numbers" because they partition the number line. Sometimes they are called "Key Numbers".)

The only places that the expression on the left **might** change sign are

The partitions of the number line are (the intervals):

We need to test each interval. There are a few ways to do this, but for many students, the simplest is to pick a test number in each interval and determine the sign of the expression.

To solve:

The solutions are all the numbers in

Using graphing technology:

#### Explanation:

To find the solutions of

graph{y = (x-1)/(2x+3) -1 [-10.65, 9.36, -4.26, 5.74]}

You can zoom in with a mouse wheel and you can hold and drag the graph around as needed.

It is, I think, fairly clear that the graph is below the

It is also clear that from

That other negative

So the solutions are all values of