1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
vlabodo [156]
2 years ago
15

Solve the given inequality :

Mathematics
1 answer:
tigry1 [53]2 years ago
4 0

Answer:

x < -5  or  x = 1  or  2 < x < 3  or  x > 3

Step-by-step explanation:

Given <u>rational inequality</u>:

\dfrac{(x-1)^2(x-2)^3}{(x^2-5x+6)^2(x+5)}\geq 0

\textsf{Factor }(x^2-5x+6):

\implies x^2-2x-3x+6

\implies x(x-2)-3(x-2)

\implies (x-3)(x-2)

Therefore:

\dfrac{(x-1)^2(x-2)^3}{(x-3)^2(x-2)^2(x+5)}\geq 0

Find the roots by solving f(x) = 0  (set the numerator to zero):

\implies (x-1)^2(x-2)^3=0

\implies (x-1)^2=0\implies x=1

\implies (x-2)^3=0 \implies x=2

Find the restrictions by solving f(x) = <em>undefined  </em>(set the denominator to zero):

\implies (x-3)^2(x-2)^2(x+5)=0

\implies (x-3)^2=0 \implies x=3

\implies (x-2)^2=0 \implies x=2

\implies (x+5)=0 \implies x=-5

Create a sign chart, using closed dots for the <u>roots</u> and open dots for the <u>restrictions</u> (see attached).

Choose a test value for each region, including one to the left of all the critical values and one to the right of all the critical values.

Test values:  -6, 0, 1.5, 2.5, 4

For each test value, determine if the function is positive or negative:

f(-6)=\dfrac{(-6-1)^2(-6-2)^3}{(-6-3)^2(-6-2)^2(-6+5)}=\dfrac{(+)(-)}{(+)(+)(-)}=+

f(0)=\dfrac{(0-1)^2(0-2)^3}{(0-3)^2(0-2)^2(0+5)}=\dfrac{(+)(-)}{(+)(+)(+)}=-

f(1.5)=\dfrac{(1.5-1)^2(1.5-2)^3}{(1.5-3)^2(1.5-2)^2(1.5+5)}=\dfrac{(+)(-)}{(+)(+)(+)}=-

f(2.5)=\dfrac{(2.5-1)^2(2.5-2)^3}{(2.5-3)^2(2.5-2)^2(2.5+5)}=\dfrac{(+)(+)}{(+)(+)(+)}=+

f(4)=\dfrac{(4-1)^2(4-2)^3}{(4-3)^2(4-2)^2(4+5)}=\dfrac{(+)(+)}{(+)(+)(+)}=+

Record the results on the sign chart for each region (see attached).

As we need to find the values for which f(x) ≥ 0, shade the appropriate regions (zero or positive) on the sign chart (see attached).

Therefore, the solution set is:

x < -5  or  x = 1  or  2 < x < 3  or  x > 3

As interval notation:

(- \infty,-5) \cup x=1 \cup (2,3) \cup(3,\infty)

You might be interested in
Dean needs $150 for a miming class. He already has $85 and he can earn the rest by working 13 hours at his job. If h represents
AnnyKZ [126]

Answer:

h=$65/13

Step-by-step explanation:

So he has $85

Now the remaining amount he needs is $65

And in every 13h he produces the $65

So to find his hourly earning is h=$65/13

So the equation is

h = $65/13

3 0
3 years ago
Need help combining like terms
yaroslaw [1]

Answer:

-10x+7.5

Step-by-step explanation:

-2.5(4x-3)

Multiply each term by -2.5

-2.5 * 4x = -10x

-2.5 * 3 = 7.5

-10x + 7.5

7 0
2 years ago
A local University accepted 2,300 students out of 4,500 applicants for admission. What was the acceptance rate, expressed as a p
Mademuasel [1]
2300 out of 4500 were chosen

2300 / 4500 = 0.5111

0.51 = 51%

51% is the acceptance rate.
4 0
3 years ago
Read 2 more answers
Which pair of angles are congruent or supplementary
Fittoniya [83]
Angles 1 and 3 are supplementary
6 0
2 years ago
What is 1/2 times 6?
son4ous [18]
1/2 times 6 = 1/2 * 6
so in order to make this multiplication you have to turn 6 to a fraction, 6 is equal to 6/1
so , the way to multiply them is multiply the top with the top and it will be the top in the answer and the exact same with the bottoms

6*1= 6
2*1=2 

so the fraction is 6/2, since a fraction is basicly a division

6/2 = 3

so the answer to your question is 3

i hope i helped you
6 0
3 years ago
Read 2 more answers
Other questions:
  • Question
    13·2 answers
  • Jill bought a pound of strawberries for $4.00. What is the price, in dollars, per ounce of strawberries? (1lb = 16 oz)
    10·2 answers
  • There are 3 students to 1 adult on a field trip.You need to find the number of adults needed for 24 students. What would the sec
    12·1 answer
  • Drew has 50 red,green,and blue marbles. he has six more red marbles than blue marbles and four fewer green marbles than blue mar
    8·1 answer
  • Math Question Who ever helps gets brainlist
    14·1 answer
  • Evaluate the expression for the given value of the variable
    11·1 answer
  • Is anyone good at Simplifying Radical Expressions?
    5·1 answer
  • What is the value of the function y=−2x+11 when x = 5?
    14·2 answers
  • 3.
    13·1 answer
  • here is my question: Find the rule which gives the number of matchsticks required to make the following matchstick patterns?
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!