Rational Inequality...6

[nycmathguy]

Section 2.7
Question 52

Note: For question 52, most of the math work was done on paper. I will show the highlights.

3x/(x - 1) ≤ x/(x + 4) + 3

This can be simplified to become

(-x^2 + 4x + 12)/((x - 1)(x + 4) ≤ 0

Setting the numerator = 0 we get x = -2, x = 6.
Setting denominator = 0 we get x = -4, x = 1.

This is what the values of x look like on the real number line:

When x = -5, we get -11/2 ≤ 0. True statement.
When x = -3, we get 9/4 ≤ 0. False statement.
When x = -2, we get 0 ≤ 0. True statement..
When x = -1, we get 7/6 ≤ 0. True statement.
When x = 2, we get 8/3 ≤ 0. False statement.
When x = 6, we get 0 ≤ 0. True statement.
When x = 7, we get -3/22 ≤ 0. True statement.

Here are the intervals that satisfy the original rational inequality:

(-infinity, -4) U [-2, 1) U [6, infinity)

On the real number line it looks like this:

You say?

 

[MathLover1]

perfect

here is the graph

 

[nycmathguy]

perfect

here is the graph

Weird-looking graph. All rational inequality graphs have a weird look on the xy-plane. As a math person, I like all graphs.

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