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first simplify the numerator (x+3)(x-2)=x^2-2x+3x-6=x^2+x-6
now factor the numerator (x+3)(x-2)
now find the restriction 3x-6=0 3x=6 x=2
now factor the denominator 3(x-2)
((x+3)(x-2))/3(x-2)
now cross off the (x-2)
and you are left with (x+3)/3
ANSWERS:
The restriction is X=2
The simplified fraction is (x+3)/3
![\frac{x-7}{x+4} / \frac{x}{x+3} = \frac{x-7}{x+4} * \frac{x+3}{x} = \frac{(x-7)(x+3)}{(x+4)(x)}](https://tex.z-dn.net/?f=%20%20%5Cfrac%7Bx-7%7D%7Bx%2B4%7D%20%2F%20%5Cfrac%7Bx%7D%7Bx%2B3%7D%20%3D%20%20%5Cfrac%7Bx-7%7D%7Bx%2B4%7D%20%2A%20%5Cfrac%7Bx%2B3%7D%7Bx%7D%20%3D%20%20%5Cfrac%7B%28x-7%29%28x%2B3%29%7D%7B%28x%2B4%29%28x%29%7D%20)
Remember that dividing by a fraction is the same as multiplying by the reciprocal (or the reverse) of the fraction on the bottom. A restriction on the domain means that x can not be that value. This occurs when the denominator is zero. If (x+4) = 0 or x = o then the denominator will be zero so the restrictions are x = -4,0. The answer is c.
Answers:
- Distributive Property
- Inverse Property
- Identity Property
- Associative Property
- Commutative Property
- Multiplication Property of Zero
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Explanations:
- The distributive property is a*(b+c) = a*b+a*c. We multiply the outer term 'a' by each term inside (b and c), then add up the results. In this specific case, we are multiplying the outer 2 by x and 3. So that's why 2(x+3) = 2x+2*3 = 2x+6. The concept of factoring takes this process in reverse, so we go from 2x+6 to 2(x+3).
- The inverse property, specifically the additive inverse property, is where we can add any number to its negative counterpart to always get 0. The expression 2+(-2) is the same as 2-2. We can think of it like "we're on the 2nd floor and we go down 2 floors to end up on floor 0". In general, the additive inverse property is x+(-x) = 0, which is the same as -x+x = 0.
- We can multiply any number by 1, to get the same number. So that's why 1*x = 1x = x. Similarly, x*1 = x as well. This is the multiplicative identity property, often shortened to "identity property".
- The parenthesis shifted around, so this means we'll use the associative property. In general, that is a+(b+c) = (a+b)+c.
- We can multiply two numbers in any order. The general format is a*b = b*a. This is the commutative property of multiplication. The version for addition is a+b = b+a.
- Multiplying 0 by any number leads to 0. So we could have the most complicated expression thought possible, but if we multiply it by 0, then the whole thing goes to 0. At the end of this complicated expression is where the 0 is buried. This idea is useful when it comes to the zero product property where if A*B = 0, then either A = 0 or B = 0 or both are the case.