Question #22417

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

Expert's answer

15a. 2x^2 - 5x<= 3

2x^2 - 5x -3<= 0

Let us solve the equation

2x^2 - 5x -3 =0

D = 25 + 4 * 2* 3 = 49 = 7^2

x1 =(5-7)/(2*2) = -2/4 = -1/2 = -0.5

x2 =(5+7)/(2*2) = 12/4 = 3

As the coefficient at x^2 is 2>0, the branches of theparabola directed upward, and so the solution of the eqaution is

[-0.5, 3]

Answer: [-0.5, 3]

-------------------*=============================*-------------------->

-0.5 3

the ends are included, and the solution is shown with====

-------------------------------------------------

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

This inequality reduces to the following system:

(x+3)(x-4)>0

and

x-4 <> 0

The solution of the first inequality:

(x+3)(x-4)>0

x in(-infinity, -3) U (4, +infinity)

Since this set does not contain 4, it is a solution ofthe initial system

Answer: (-infinity, -3) U (4, +infinity)

================o---------------------o===============================>

-3 4

the ends are not included, and the solution is shown with====

2x^2 - 5x -3<= 0

Let us solve the equation

2x^2 - 5x -3 =0

D = 25 + 4 * 2* 3 = 49 = 7^2

x1 =(5-7)/(2*2) = -2/4 = -1/2 = -0.5

x2 =(5+7)/(2*2) = 12/4 = 3

As the coefficient at x^2 is 2>0, the branches of theparabola directed upward, and so the solution of the eqaution is

[-0.5, 3]

Answer: [-0.5, 3]

-------------------*=============================*-------------------->

-0.5 3

the ends are included, and the solution is shown with====

-------------------------------------------------

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

This inequality reduces to the following system:

(x+3)(x-4)>0

and

x-4 <> 0

The solution of the first inequality:

(x+3)(x-4)>0

x in(-infinity, -3) U (4, +infinity)

Since this set does not contain 4, it is a solution ofthe initial system

Answer: (-infinity, -3) U (4, +infinity)

================o---------------------o===============================>

-3 4

the ends are not included, and the solution is shown with====

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