Analyze the solution shown:

1. [tex]\[ -|-x| = 7 \][/tex] (given)
2. [tex]\[ |-x| = -7 \][/tex] (multiplication property of equality)
3. [tex]\[ -x = 7 \][/tex] or [tex]\[ -x = -7 \][/tex] (definition of absolute value)
4. [tex]\[ x = -7 \][/tex] or [tex]\[ x = 7 \][/tex] (multiplication property of equality)
5. Check:
[tex]\[ -1-(-7) \mid = 7, -7 \neq 7 \][/tex]
[tex]\[ -|-7| = 7, 7 = 7 \][/tex]

Negative times negative is positive. Determine the flaws in the solution.

A. The multiplication property of equality in Step 2 should result in [tex]\[ |x| = 7 \][/tex].
B. The definition of absolute value is applied incorrectly. There is no solution.
C. The reason in Step 4 should be the addition property of equality.
D. The multiplication property of equality in Step 4 should result in only [tex]\[ x = 7 \][/tex].
E. In evaluating the solutions, the absolute value should be simplified first to get a positive value.



Answer :

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.