Answer to Question #22342 in Trigonometry for Kristen Woods

Question

Solve, write your answer in interval notation and graph the solution set.

15a. 2x^2-5x less than or equal to 3

15b. (x+3)/(x-4) greater than 0

Answer 1

2x^2 - 5x <=3
First solve the equation 2x^2 - 5x = 3
2x^2 - 5x - 3 =0
D = 25 + 4*2*3= 49 = 7^2
x1 =(5-7)/(2*2) = -2/4 = -0.5
x2 =(5+7)/(2*2) = 12/4 = 3
As the coefficient at x^2 is 2>0 he branches of theparabola
y = 2x^2 - 5x -3
are directed upwards, whence the inequality
2x^2 - 5x <=3
Has the following solution
x belongs to[-0.5, 3]
Solution set:
--------*===========================*--------------->
-0.5 3

Here ===== means solution interval,
and * means that the corresponding point belogns to thatinterval15b. (x+3)/(x-4) greater than 0
Solution:
(x+3)/(x-4)> 0
This inequality is equivalent to the following system:
(x+3)(x-4)>0
x-4 <>0 ( <> means not equal )
(x+3)(x-4)>0
x <>4 ( <> means not equal )
This first inequality has the solution
x in (-3,4)
and this does not contains the point 4, so
(-3,4)
is the solution of the initial inequality
Solution set: --------o====================o--------------->
-3 4

Here ===== means solution interval,
and o means that the corresponding point does not belongto that interval