To solve the equation [tex]\( -2|-x| + 5 = 8 \)[/tex], follow these steps:
1. Isolate the absolute value expression:
We begin by isolating the term involving the absolute value. Start by subtracting 5 from both sides of the equation:
[tex]\[
-2|-x| + 5 - 5 = 8 - 5
\][/tex]
Simplifying this, we get:
[tex]\[
-2|-x| = 3
\][/tex]
2. Solve for the absolute value term:
Next, divide both sides of the equation by -2 to isolate [tex]\( |-x| \)[/tex]:
[tex]\[
\frac{-2|-x|}{-2} = \frac{3}{-2}
\][/tex]
Simplifying this, we get:
[tex]\[
|-x| = -\frac{3}{2}
\][/tex]
3. Analyze the absolute value result:
The absolute value of any expression is always non-negative (i.e., it is always greater than or equal to 0). However, we have:
[tex]\[
|-x| = -\frac{3}{2}
\][/tex]
Since [tex]\( -\frac{3}{2} \)[/tex] is negative, it is impossible for the absolute value of any real number to equal a negative number.
Therefore, there are no solutions to the equation [tex]\( -2|-x| + 5 = 8 \)[/tex]. This means the equation has no real solutions.
