Solve Inequality Algebraically

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harpazo

harpazo

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Solve the inequality algebraically.

(x - 5) (x + 2)^2 > 0

It's been a while since I did one of these. Seeking steps. Textbook instructions are not too clear.
 
harpazo said:
Solve the inequality algebraically.
(x - 5) (x + 2)^2 > 0
Click to expand...
Note that for all \(\displaystyle x\) the factor \(\displaystyle (x+2)^2\ge 0\).
So with the exception of \(\displaystyle x=-2\) all depends on the factor \(\displaystyle (x-5)\)
 
pka said:
Note that for all \(\displaystyle x\) the factor \(\displaystyle (x+2)^2\ge 0\).
So with the exception of \(\displaystyle x=-2\) all depends on the factor \(\displaystyle (x-5)\)
Click to expand...

What???
 
harpazo said:
Parity???
Click to expand...

The parity of a number refers to whether it is odd or even. The behavior of a function near a zero, will be different depending on the parity of the zero. If the parity is odd, the function will change sign as it moves across the zero, if the parity is even, it will not change sign, that is, the function will be tangent to the horizontal axis there.
 
(x - 5) (x + 2)^2 > 0

x - 5 > 0

x > 5

(x + 2)^2 > 0

sqrt{(x + 2)^2} > sqrt{0}

x + 2 > 0

x > -2

<---------(-2)----------------(5)---------->

Pick a number from each interval to evaluate in the given inequality.

Letting x = -3 leads to a false statement.

(x - 5) (x + 2)^2 > 0

Letting x = 0 leads to a false statement.

Letting x = 6 leads to a true statement.

Answer: (5, infinity).

We can also express the answer as x > 5.
 
harpazo said:
What are you talking about?
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When you reply to a post with the single word "what", it only tells us that you're confused about something. As tutors, we need to know what that something is.

Please be specific, when you ask a question. That way, tutors will know which part(s) you're asking about.

I hope this makes sense for you, now, but if you're still confused over what I'm talking about, please feel free to say so and also try to specify the word or phrase where you're stuck. Thank you.

\(\;\)
 
Otis said:
When you reply to a post with the single word "what", it only tells us that you're confused about something. As tutors, we need to know what that something is.

Please be specific, when you ask a question. That way, tutors will know which part(s) you're asking about.

I hope this makes sense for you, now, but if you're still confused over what I'm talking about, please feel free to say so and also try to specify the word or phrase where you're stuck. Thank you.

\(\;\)
Click to expand...

Ok. Understood.
 
(x - 5) (x + 2)^2 > 0

On the left-hand side, we see a product of two numbers. One is x-5 and the other is the square of x+2.

(x + 2)^2 is never negative because it's a square.

When the product (x - 5)(x + 2)^2 is greater than zero, then it is positive.

So, we can see that (x - 5) must be positive.

positive × positive > 0

(If x-5 were not positive, then the product would not be greater than zero.)

Therefore, it's clear that:

x - 5 > 0

So the solution is x > 5

That approach in thinking is what pka was hinting at, in post #3.

?
 
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