Specific factoring question

In summary, the conversation discusses the problem of 2(x+3)(x-2)/(x-2)≥0 and why it is incorrect to factor x-2 in the numerator against x-2 in the denominator. It is mentioned that this is because the inequality does not have a solution when x=2 due to division by 0. The conversation also touches on the importance of not dividing or multiplying by the denominator on both sides of the inequality sign, as it can lead to uncertainty and potentially incorrect solutions.
  • #1
LogarithmLuke
83
3
So given the problem 2(x+3)(x-2)/(x-2)≥0

How come we can't factor x-2 in the numerator against x-2 in the denominator? I mean theyre all just factors right? Does it have something to do about it being an inequality? We got this problem on a test yesterday, and the correct way to solve it was to leave it as i originally wrote it, and then put it on a number line. Why is it wrong to just factor and solve the inequality from there?
 
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  • #2
It's because that the inequality ##\frac{2(x+3)(x-2)}{x-2}\ge 0## doesn't have the solution ##x=2## for the denominator shouldn't be ##0.## But, the inequality ##2(x+3)\ge 0## does.
So, if you want to divide something in numerator and denominator simultaneously, you should confirm it would not matter its original solution, that is, think clearly.
 
  • #3
What you call factoring is in fact division by x-2. This can only be done if x is not equal to 2 (you can not divide by 0, not even when you do it both on top and on the bottom). If you solve x+3 > 0 you get x > -3 and have missed excluding this x=2 for the set of x that satisfies the original inequality.

[edit] slow typist. Good thing the replies agree.
 
  • #4
Oh i see now. I am aware that you can't divide or multiply by the denominator on both sides of the inequality sign, because you can't know if you have to swap the inequality sign or not. I just wasnt aware that i was doing that here.
 
  • #5
LogarithmLuke said:
Oh i see now. I am aware that you can't divide or multiply by the denominator on both sides of the inequality sign, because you can't know if you have to swap the inequality sign or not. I just wasnt aware that i was doing that here.
Do I understand this ? you want to change ##2(x+3)(x-2)/(x-2)\ge 0## into ##2(x+3)(x-2) \ge 0 \; (x-2) ## and worry about the ##\ge## ?
 
  • #6
No, i meant that dividing x-2 by x-2 would leave uncertainty about whether or not you have to swap the inequality sign, like you guys said.
 
  • #7
It leaves the inequality sign intact. But you can't do it if x-2 is 0, that's the crux.
 
  • #8
Oh, i was thinking about the fact that if you mulitply or divide by something negative on both sides, you have to swap the inequality sign, but i see now that that is not relevant here because you only divide on one side.
 
  • #9
OK, I'm reassured now. Well done.
 

1. What is specific factoring and why is it important?

Specific factoring is a mathematical technique used to break down an algebraic expression into smaller factors. It is important because it allows us to simplify complex expressions and solve equations more easily.

2. How do you factor a quadratic expression?

To factor a quadratic expression, we use the "AC" method or the "x-box" method. These involve finding two numbers that multiply to give the constant term (C) and add to give the coefficient of the middle term (B). These numbers are then used to create two binomial factors.

3. Can you factor expressions with more than two terms?

Yes, expressions with more than two terms can be factored using techniques such as grouping, difference of squares, and perfect square trinomials. It is important to look for common factors and patterns when factoring expressions with more than two terms.

4. How does factoring help in solving equations?

Factoring is an important step in solving equations because it allows us to transform a complex expression into a simpler form that is easier to solve. By factoring, we can isolate the variable and find its value more easily.

5. Are there any real-life applications of factoring?

Yes, factoring is used in many real-life situations, such as in finance, economics, and engineering. It can be used to simplify and solve complex equations that arise in these fields. For example, in finance, factoring is used to calculate interest rates and compound interest, while in engineering, it is used to solve problems related to forces and motion.

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