Linear Inequality

Here are some tips for dealing with inequality.
Many students arrive at the last step with something like the following.
(1) (x – a)(x – b)
(2) (x – a)(x – b) < 0 
(3) (x – a)(x – b) ≥ 0
(4) (x – a)(x – b) >  0

Method 1:
Now we assume a < b
Answer for (1) : ≤ x ≤ b
Answer for (2) : a < x < b
Answer for (3) : ≤ a or x ≥ b
Answer for (4) : x < a or x > b


Method 2 is to draw a graph. Here is an example.





Turn into (x – a)(x – b) form






































Answer : D

For this type of question, you may assign real values for a and b. For instance, a = -2, b = -1.

It is easily seen that D is the answer.
If  the condition is just a < b, you have to consider all the probabilities. 

(1) Both positive
(2) Both negative
(3.1) One negative, one positive (a = -1, b = 2)
(3.2) One negative, one positive (a = -2, b = 1)








Answer : E
6x2–7x–20 0
(2x –5)(3x + 4) 0
(x–5/2)(x + 4/3) 0
-4/3 ≤ x ≤ 5/2

Hence, we have -1, 0, 1, 2.







Answer : D

Draw a picture and find the intersection part.









空心代表 >/<
實心代表 ≥/≤

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