Answer:
1. (-10x -5) -(-7x -4)
2. (-10x -5) -(-3x -1)
Step-by-step explanation:
The process of simplifying an expression generally involves eliminating parentheses, collecting terms, simplifying fractions, rationalizing denominators, removing appropriate factors from under radicals, and generally putting expressions into general form.
So, to accomplish what the problem is asking, you can create an expression that requires you accomplish as many or as few of these steps as you may like.
In general, you will want to both "do" and "undo" whatever operation(s) you may choose. For example, consider a couple of expressions "a" and "b". For this example, we want our final result to simplify to "a".
If we choose to add "b", we might have ...
(a+b) -b . . . . . we have both added and subtracted b, so the simplified result is "a"
If we choose to multiply by "b", we might have ...
(ab)/b . . . . . . we have both multiplied by b and divided by b, so the simplified result is again "a"
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We can solve both these problems at once by using ...
If we choose the first approach, we want an expression that is (a+b). That will be ...
a+b = (-3x -1) +(-7x -4) = -10x -5
By subtracting "b" from this expression, we get "a"--and vice versa. That is ...
- (-10x -5) -(-7x -4) . . . simplifies to . . . -3x -1
- (-10x -5) -(-3x -1) . . . simplifies to . . . -7x -4
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<em>Alternate solution to 2</em>
Suppose we choose to multiply instead. Let "b" be -3x+2. Then, using the template for multiplication above, we can write the expression
(-7x -4)(-3x +2)/(-3x +2) = (21x^2 -2x -8)/(-3x +2)
and we know this rational expression simplifies to -7x -4.
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Of course, we can be as elaborate as we might like in modifying the original expression. Whatever we do, we must also include the "undo" so the original expression is recovered when we simplify. To illustrate the point, we can add and subtract 2x from the above rational expression.
((21x^2 -2x -8)/(-3x +2) + 2x) - 2x = (21x^2 -2x -8 +2x(-3x +2))/(-3x +2) - 2x
= (21x^2 -2x -8 -6x^2 +4x)/(-3x +2) -2x
= (15x^2 +2x -8)/(-3x +2) -2x . . . . . . another alternate solution to #2.
If we're not careful, we can create an expression that is extremely difficult to simplify. The one we have here can be factored to be ...
(-5x -4)(-3x +2)/(-3x +2) -2x = -5x -4 -2x = -7x -4 . . . our original #2 expression
Knowing that it <em>can</em> be factored is a big help.
