Let's analyze the provided argument step-by-step to identify the flaws and correct the reasoning where necessary.
1. Given Expression: [tex]\( -|-x| = 7 \)[/tex]
- This expression is the initial equation provided.
2. Incorrect Step 2: [tex]\(|-x| = -7\)[/tex]
- This step assumes that applying the multiplication property of equality directly results in [tex]\(|-x| = -7\)[/tex].
- Flaw: The absolute value of any real number is always non-negative. Therefore, [tex]\(|-x| = -7\)[/tex] is impossible because the left side, [tex]\(|-x|\)[/tex], cannot be a negative number.
3. Definition of Absolute Value Misapplied: [tex]\(-x = 7\)[/tex] or [tex]\(-x = -7\)[/tex]
- This step further assumes that the absolute value definition can be applied directly after the faulty logic in step 2.
- Flaw: As established, step 2 is incorrect, so any step following from it is also incorrect.
4. Combining Incorrect Results: [tex]\( x = -7 \)[/tex] or [tex]\( x = 7 \)[/tex]
- Multiplying both sides of each equation by [tex]\(-1\)[/tex], we obtain [tex]\( x = -7 \)[/tex] or [tex]\( x = 7 \)[/tex].
- Flaw: Since step 2 was incorrect, these results are not valid.
5. Solution Verification:
- The step where [tex]\( -|-(-7)| = 7 \)[/tex] and [tex]\( -|7| = 7 \)[/tex] is evaluated is incorrect because it builds on previous incorrect steps.
- Although the absolute value is applied, it again wrongly assumes prior steps were accurate.
### Correct Approach:
To solve [tex]\( -|-x| = 7 \)[/tex]:
#### Step 1:
Given equation: [tex]\( -|-x| = 7 \)[/tex]
#### Step 2:
Recognize that [tex]\( |-x| = |x| \)[/tex]. The absolute value of [tex]\(-x\)[/tex] is the same as the absolute value of [tex]\( x \)[/tex].
Therefore, rewrite the equation as:
[tex]\[ -|x| = 7 \][/tex]
#### Step 3:
Analyze the equation:
[tex]\[ -|x| = 7 \][/tex]
We observe that [tex]\( |x| \)[/tex] is always non-negative, and multiplying any non-negative number by [tex]\(-1\)[/tex] results in a non-positive (zero or negative) number.
Clearly, a negative number (or zero) cannot be equal to [tex]\( 7 \)[/tex].
#### Conclusion:
There is no real value for [tex]\( x \)[/tex] that satisfies [tex]\( -|x| = 7 \)[/tex] because:
[tex]\[ -|x| \ \text{is always non-positive, and 7 is positive.} \][/tex]
Therefore:
There is no solution to the given equation.
