((3x-1))/(x-4)>=0 solve the inequality

Answer 1

It is not actually needed to test one value from each interval - it is enough to test just one.

Say check 1000. It is obvious that the whole expression is positive.

Then move to the next interval to the left - only one expression in parenthesis changes the sign, so the whole expression changes the sign - it is now negative. Move over to the next interval - it is now positive.

This method is known as "interval" method.

But I have to agree with the previous expert - the first step is to create these intervals on a numeric line.

Answer 2

Hello Danny,

Because the quantity (x - 4) is in the denominator, x≠ 4.

The numerator quantity has a critical value, x= 1/3.

So, x = 4 or x = 1/3 cause the respective quantities to equal 0, those are the critical values of x for this rational expression.

The number line is partitioned into 3 intervals: (-∞, 1/3], [1/3, 4), (4, ∞).

The bracket at 1/3 because the numerator can = 0; the parentheses at 4 because the denominator cannot = 0.

Now we use test values of x, one from each partition - not the critical values. I suggest the values x= 0, 1, and 5. There could be many other choices (I recommend to keep them 'small and simple').

(3(0) - 1)/(0 - 4) = 4, which is positive. x values in the interval (-∞, 1/3] yield positive or zero results.

(3(1) - 1)/(1 - 4) = (-2/3). x values in the interval [1/3, 4) yield negative or zero results.

(3(5) - 1)/(5 - 4) = 14/4 or 7/2. x values in the interval (4, ∞) yield positive results.

The solution in interval form:

(-∞, 1/3] ∪ (4, ∞)

Regards,

Anita A.