Let's go through the given solution step-by-step and identify any flaws.
### Given Solution Steps:
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]$-|-(-7)|=7, -7 \pm 7$[/tex]
[tex]$-|-7|=7, 7=7$[/tex]
### Analysis of Each Step:
1. Step 1: [tex]$-|-x|=7$[/tex]
This step is given correctly, and there's no issue here.
2. Step 2: [tex]$|-x|=-7$[/tex] : multiplication property of equality
Here is the first major flaw. When isolating the absolute value, the multiplication property of equality dictates that we should multiply both sides of the given equation by -1 to remove the negative sign:
[tex]\[
-|-x| = 7 \implies |-x| = -7
\][/tex]
However, absolute values cannot be negative. So, [tex]$|-x| = -7$[/tex] is incorrect. The correct multiplication results in:
[tex]\[
|-x| = 7
\][/tex]
The correct step should have been:
[tex]\[
|-x| = 7
\][/tex]
3. Step 3: [tex]$-x=7$[/tex] or [tex]$-x=-7$[/tex] : definition of absolute value
This step assumes that [tex]$|-x|$[/tex] evaluates the definition of absolute value. From [tex]$|-x| = 7$[/tex], we should have derived:
[tex]\[
x = -7 \quad \text{or} \quad -x = 7 \implies x = -7 \quad \text{or} \quad x = 7
\][/tex]
4. Step 4: [tex]$x=-7$[/tex] or [tex]$x=7$[/tex] : multiplication property of equality
Here, the step correctly states the multiplication property of equality for absolute values. Since [tex]$|-x| = 7$[/tex], this correctly translates to:
[tex]\[
x = -7 \quad \text{or} \quad x = 7
\][/tex]
5. Step 5: Check [tex]$-|-(-7)|=7,-7 \pm 7$[/tex]: [tex]$-|-7|=7, 7=7$[/tex]
The check process seems correct when plugging in the values [tex]$x = -7$[/tex] and [tex]$x = 7$[/tex] back into the original equation:
[tex]\[
-|-(-7)| = 7 \rightarrow -|-7| = 7 \rightarrow -7 = 7 \quad \text{True}
\][/tex]
For [tex]$x=7$[/tex]:
[tex]\[
-|-(7)| = -|-7| = -7 = 7 \quad \text{True}
\][/tex]
### Summary of Flaws:
1. Step 2 Miscalculation:
The multiplication property of equality should result in:
[tex]\[
|x| = 7 \quad \text{and not} \quad |-x| = -7
\][/tex]
2. Definition of Absolute Value Applied Correctly:
The original application after this correction would indeed lead towards correctly finding [tex]$x = 7$[/tex] or [tex]$x = -7$[/tex].
3. Addition Property Mislabelled:
Multiplication property is misapplied here; instead, solving the expressions should be listed under suitable algebra aa suchthis leads the correct roots.
By correcting Step 2 as above and verifying results, we impose:
- The error in the solution is primarily Step 2 where the multiplication property should yield [tex]$|x|=7$[/tex].
Thus, ensuring the solution process is consistent and correctly solves the expression for the roots.
