Answer:
A
Explanation:
if x < 0, then by the definition of the absolute value function, |x| > 0. this is because the absolute value function turns whatever is inside the function, whether positive or negative, into positive.
also, the product of a negative number and a positive number is negative.
let's do them step by step.
a) x|x|<0 is the product of a negative value (x) and a positive value (|x|). as mentioned earlier the product of the following is negative. thus, this is true.
b) x|x|²>0 is again, the product of a negative value (x) and a positive value (|x|², since the square of any value is positive). so the product is negative. thus, this statement is false.
c) x|x|>0 is false, as established from (a).
d) note that here x is the negative of |x| so essentially it is the additive inverse of |x|. by definition the sum of a number and it's additive inverse is equal to 0. (additive inverse of n is -n).
so, we have x+|x|=0, however the statement is false since it is inequality, not equality. it will never take values greater than 0.
e) lastly, -|x| can be expressed simply as x, so we have the sum 2x>0, which is false as x<0.
therefore, our only solution is (a)