Answer: There are no solutions.
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Explanation:
Given the equation:
4x + 2(x − 5) = 3(2x − 4) ;
How many solutions for x exist?
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Note: from the left-hand side of the equation ^given:
+ 2(x − 5) ;
Let's simplify this part:
Note the distributive property of multiplication:
a (b + c) = ab + ac ;
So: +2 ((x−5) = (2*x)) + (2*-5) ;
= 2x + (-10) ;
= 2x − 5 ;
Now, bring down the 4x from the left-hand side of the equation:
4x + (2x − 5) ;
Rewrite as: 4x + 2x − 5 ;
and simplify by combining the like terms :
4x + 2x = 6x ;
and bring down the -5 ; to rewrite the left-hand side of the equation:
6x − 5 ;
Now, for the right-hand side of the equation, we have:
3(2x − 4) ;
Let us simplify:
Again. Note the distributive property of multiplication:
a (b + c) = ab + ac ;
So: +3 ((2x−4) = (3*2x)) + (3 * -4) ;
= 6x + (-12) ;
= 6x − 12 ; for the right-hand side of the equation.
Now: Combine both simplified expressed from the left-hand and the right-hand sides :
6x − 5 = 6x − 12 ;
If we subtract 6x from each side of the equation,
6x − 5 − 6x = 6x − 12 − 6x ;
we are left with -5 = - 12 ; which is not true.
As such, for the "equation" given—
4x + 2(x−5)=3(2x −4) ; there are no solutions.
This is sometimes called a null or empty set— or { } .