(3 x)/(x+5) ÷(6)/(4 x+20) Which of the following is equivalent to the expression above, given th

step 1 Okay, so this question, now there's a couple routes through it. The first thing I want to take n

(3 x)/(x+5) ÷(6)/(4 x+20) Which of the following is...

Key Concepts

Domain Restrictions

Domain restrictions are conditions that specify the values for which a rational expression is defined, usually by setting denominators not equal to zero. Identifying and applying these restrictions is important to avoid division by zero and ensure the expression is valid for the given values.

Factoring

Factoring is the process of rewriting an expression as a product of its factors. In the context of rational expressions, factoring is critical because it can reveal common factors between the numerator and the denominator, which can then be canceled to simplify the expression.

Simplification of Rational Expressions

Simplifying rational expressions involves combining fractions through multiplication or division and then reducing the result by canceling out common factors present in both the numerator and the denominator. This process often requires careful consideration of restrictions to ensure that variables are within allowed values.

Division of Fractions

Division of fractions involves replacing the division operation with multiplication by the reciprocal of the divisor. This means that when one fraction is divided by another, you invert (or swap) the numerator and the denominator of the second fraction and then multiply the two fractions together.

Transcript

00:01 Okay, so this question, now there's a couple routes through it.

00:07 The first thing i want to take note of is don't be wigged out by this x does not equal negative five.

00:13 Any of these little add -on caveats at the end of the thing are often just them covering their bases.

00:20 In this case, they're covering their bases to say x cannot equal negative five because if it did, the denominator would be zero, which is defined.

00:31 So you don't have to make anything of that x doesn't equal negative 5.

00:34 It's just them covering their basis.

00:35 So don't be wigged out by that little detail.

00:38 Another thing is like if they're like x is a constant, don't be wigged out by that.

00:42 It's just them covering their bases.

00:44 Now there are two ways through this.

00:46 The first is the algebraic way and the second is the strategic way.

00:50 I'll start with the algebraic.

00:53 What's difficult here is they are banking on you being intimidated by dividing fractions, but remember that dividing fractions is the same thing as multiplying by the reciprocal.

01:06 So i can rewrite this using multiplication as 3x over x plus 5 times the reciprocal, just meaning flip the fraction.

01:16 So times 4x plus 20 over 6.

01:21 And now multiplying, multiplying fractions, i just multiply straight across numerator by numerator and denominator by denominator.

01:29 So i'll put that here plus 20.

01:33 This is just so you can see x plus 5 times 6.

01:39 So now i can see clearly what that is.

01:41 And then i'll keep going straight across here.

01:45 Distribute.

01:46 So 3x times 4x is 12x squared.

01:50 Distribute second term.

01:51 3x times 20.

01:54 Positive 20.

01:55 Mind your signage here is 60x.

01:58 Cool.

01:59 Over denomination same thing x times 6x 5 times 6 is 30 plus 30 great i'm gonna keep sliding down here so now we're in this place we're close to it we just need to simplify i notice that there's more to be done in the numerator i can pull out they have a 12 in common and they have an x in common factor that out so this is like the this is the opposite of what we did actually.

02:34 Now, what's interesting about that is, is factoring and distributing are the opposites of each other.

02:39 So i just distributed and now i'm going to factor.

02:44 I suppose that's not the only way to go about it.

02:47 There are lots of ways.

02:47 What we're looking is to do is just to kind of simplify this as much as possible.

02:52 I think it's easy to think about things in terms of this was an obvious case for distributing and this will be an obvious case for factoring.

03:02 Whatever gets you there.

03:03 So we're factoring out the 12x left in the parentheses would be x plus 5 because i can think about going backwards.

03:09 12x times x is x squared and 12x times 5 is 60.

03:13 So it's just the opposite.

03:14 Over, over here i can pull out a 6.

03:16 6 times x plus 5 as well.

03:19 And this is the beauty part that we wanted to happen.

03:22 This often happens in questions like this.

03:24 The x plus 5 is cancel because x plus 5 divided by x plus 5 is just 1.

03:28 So that gets to go away.

03:29 Now, 12 and 6 have something in common.

03:32 I can take out a 6 from both of those, so the 6 becomes 1, the 12 becomes 2, and the x stays put.

03:40 So the final version of this is just 2x or 2x over 1.

03:44 So that would lead us up here to this answer choice.

03:48 Now, the second thing i want to demonstrate here is the strategic element.

03:54 There's this strategy for dealing with math questions called substitution.

03:59 Let me grab this.

04:02 Substitution, which you can use when there are variables in both the question and in the answers.

04:15 This tells you that the variables are not meant to be solved for...

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