# Answer to Question #22342 in Trigonometry for Kristen Woods

Question #22342

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

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

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

Expert's answer

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

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

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