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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)

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Maru [420]

Answer:

\Huge \boxed{X

Step-by-step explanation:

To solve this problem, first you have to isolate it on one side of the equation. Remember to solve this problem, find represents the solution of 5(x+5)<85.

First, you divide by 5 from both sides.

\displaystyle \frac{5(x+5)}{5}

Solve.

\displaystyle 85\div5=17

\displaystyle x+5

Next, subtract 5 from both sides.

\displaystyle x+5-5

Solve.

\displaystyle 17-5=12

As a result, the correct answer is x<12.

<h2>Hope this helps! </h2><h2 /><h2>Have a wonderful blessing day! :)</h2><h2 /><h2>Good luck! :)</h2>
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Step-by-step explanation:

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Anyone can help me solve this equation using cross multiplying
Natali5045456 [20]

9514 1404 393

Answer:

  x = 1 or 5

Step-by-step explanation:

The notion of "cross-multiplying" is the idea that the numerator on the left is multiplied by the denominator on the right, and the numerator on the right is multiplied by the denominator on the left. This looks like ...

  \displaystyle \frac{x-1}{7}=\frac{2x-2}{3x-1}\ \longrightarrow\ (x-1)(3x-1)=(7)(2x-2)

Then the solution proceeds by eliminating parentheses, and solving the resulting quadratic equation.

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_____

<em>Comment on "cross multiply"</em>

Like a lot of instructions in Algebra courses, the idea of "cross multiply" describes <em>what the result looks like</em>. It doesn't adequately describe how you get there. The <em>one and only rule</em> in solving Algebra problems is "<em>whatever is done to one side of the equation must also be done to the other side of the equation</em>." If you multiply one side by one thing and the other side by a different thing, you are violating this rule.

What looks like "cross multiply" is really "<em>multiply by the product of the denominators</em> and cancel like terms from numerator and denominator." Here's what that looks like with the intermediate steps added.

  \displaystyle \frac{x-1}{7}=\frac{2x-2}{3x-1}\\\\\frac{x-1}{7}\times7(3x-1)=\frac{2x-2}{3x-1}\times7(3x-1)\\\\(x-1)(3x-1)=(2x-2)(7)\qquad\textit{looks like}\text{ cross multiply}

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A leading automobile magazine has 60 pages and 4,50,000 copies are printed every month. How many pages in all are printed in the
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Answer:

See the explanation below

Step-by-step explanation:

Number of months =4

Number of copies =450000

Number of pages =60

Number of copies in 4 months =450000*4

=1800000

Number of pages is

=1800000 *60

=108,000,000

7 0
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