X+y=13 and 1/2x+y=10
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I agree with what that person told you. The expression -2(x+10) is equivalent to -2x-20
Multiply the outer -2 with the inner x to get -2 times x = -2x
Multiply the outer -2 with the inner term 10 to get -2 times 10 = -20
So overall, -2(x+10) = -2x-20
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So that's how we're able to go from this
to this 
The only thing that changes is the left side. The inequality sign will not flip. It only flips if we multiply both sides by a negative number.
Multiply the outer -2 with the inner x to get -2 times x = -2x
Multiply the outer -2 with the inner term 10 to get -2 times 10 = -20
So overall, -2(x+10) = -2x-20
--------------------------------------------
So that's how we're able to go from this
The only thing that changes is the left side. The inequality sign will not flip. It only flips if we multiply both sides by a negative number.