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Chapter 0: Problem 154

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$-x-x=-x+(-x)=0$$

Short Answer

Expert verified
The original mathematical statement \(-x - x = -x + (-x) = 0\) is partially true. While \(-x - x\) does equal to \(-x + (-x)\), it doesn't necessarily equal to zero. Therefore, the corrected statement should be \(-x - x = -x + (-x) = 0, only when x = 0\).

Step by step solution

01

Evaluate the Left-Hand Side

First, let's evaluate the left-hand side of the equality. According to the properties of negative numbers, \(-x - x\) can be rewritten as \(-1x + (-1x)\), which simplifies to \(-2x\).
02

Evaluate the Right-Hand Side

Now, evaluate the right-hand side of the equality. The expression \(-x + (-x)\) is similar to the left side and thus simplifies to \(-2x\) as well.
03

Compare Both Sides of the Equality

Having evaluated both sides of the equality individually, now they need to be compared. As both sides equal to \(-2x\), the given mathematical statement is true upto this point.
04

Explore the Last Part of the Statement

Now, explore the last part of the statement which suggests that \(-2x = 0\). This is a false statement because \(-2x = 0\) implies that \(x = 0\), but there was no such restriction on \(x\) in the original statement.
05

Correct the Incorrect Part of the Statement

To make the last part true, we can adjust the original statement with a qualification that \(x\) needs to be zero for the entire statement to be true. So -x-x=-x+(-x)=0 only when \(x = 0\).

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