Problem 154 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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 1: 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 \(-2x\).

Step 2: 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.

Step 3: 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.

Step 4: 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.

Step 5: 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 \(x = 0\).