Let's carefully analyze and correct the solution step by step.
### Given Equation
The problem starts with the given equation:
[tex]\[
-|-x| = 7
\][/tex]
### Step-by-Step Corrections:
#### Step 1: Start with the given equation:
[tex]\[
-|-x| = 7
\][/tex]
#### Step 2: Interpret the absolute value correctly
Recall that the absolute value of any expression is always non-negative. This means that any expression inside the absolute value operator will result in a non-negative value. Thus, [tex]\(|-x|\)[/tex] is just [tex]\(|x|\)[/tex] because the absolute value negates the negative sign.
This corrects the initial equation to:
[tex]\[
-|x| = 7
\][/tex]
However, this is problematic because the absolute value is always non-negative ([tex]\(|x| \geq 0\)[/tex]), but here it is preceded by a negative sign and set equal to a positive number (7). There's an inherent contradiction. Therefore:
[tex]\[
-|x| = 7
\][/tex]
implies:
[tex]\[
|x| = -7
\][/tex]
which is impossible because [tex]\(|x|\)[/tex] cannot be negative.
#### Correction:
Since the expression [tex]\(-|x| = 7\)[/tex] is invalid as it violates the properties of absolute values, therefore no absolute value can make the equation true.
#### Conclusion:
No such [tex]\(x\)[/tex] exists that satisfies the equation [tex]\(-|-x| = 7\)[/tex].
### Summary:
The original steps contained fundamental errors in handling the absolute value property. Specifically:
1. Rewriting [tex]\(|-x|\)[/tex] as [tex]\(|x|\)[/tex] was correct.
2. Considering that [tex]\(-|x| = 7\)[/tex] leads to an impossible result because [tex]\(|x|\)[/tex] must be non-negative [tex]\((|x| \geq 0)\)[/tex], but the given equation leads to [tex]\(|x| = -7\)[/tex], which is not possible.
### Final Answer:
The equation [tex]\(-|-x| = 7\)[/tex] has no solution.
