Let's solve each of the given equations to determine the number of solutions for each one, and then order them from least to greatest number of solutions.
1. For the equation [tex]\(\frac{1}{2}|x| + 3 = 3\)[/tex]:
[tex]\[
\frac{1}{2}|x| + 3 = 3
\][/tex]
Subtract 3 from both sides:
[tex]\[
\frac{1}{2}|x| = 0
\][/tex]
Multiply both sides by 2:
[tex]\[
|x| = 0
\][/tex]
The only value of [tex]\(x\)[/tex] that satisfies this is [tex]\(x = 0\)[/tex].
Number of solutions: 1
2. For the equation [tex]\(3 - |x + 4| = 10\)[/tex]:
[tex]\[
3 - |x + 4| = 10
\][/tex]
Subtract 3 from both sides:
[tex]\[
-|x + 4| = 7
\][/tex]
Divide by -1:
[tex]\[
|x + 4| = -7
\][/tex]
Since the absolute value cannot be negative, there are no solutions for this equation.
Number of solutions: 0
3. For the equation [tex]\(|x - 8| - 4 = -1\)[/tex]:
[tex]\[
|x - 8| - 4 = -1
\][/tex]
Add 4 to both sides:
[tex]\[
|x - 8| = 3
\][/tex]
This absolute value equation gives us two cases:
[tex]\[
x - 8 = 3 \quad \text{or} \quad x - 8 = -3
\][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[
x = 11 \quad \text{or} \quad x = 5
\][/tex]
Number of solutions: 2
Now, order the equations from least to greatest number of solutions:
1. [tex]\(3 - |x + 4| = 10\)[/tex] (0 solutions)
2. [tex]\(\frac{1}{2}|x| + 3 = 3\)[/tex] (1 solution)
3. [tex]\(|x - 8| - 4 = -1\)[/tex] (2 solutions)
Thus, the order is:
[tex]\[
3 - |x + 4| = 10 \quad \frac{1}{2}|x| + 3 = 3 \quad |x - 8| - 4 = -1
\][/tex]
