Let's solve each of the equations step-by-step and determine the number of solutions for each.
### Equation 1: [tex]\( |x-8|-4=-1 \)[/tex]
We start with:
[tex]\[ |x-8| - 4 = -1 \][/tex]
To eliminate the constant term, add 4 to both sides:
[tex]\[ |x-8| = 3 \][/tex]
Now, we have an equation involving the absolute value. The absolute value equation:
[tex]\[ |x-8| = 3 \][/tex]
has two solutions:
[tex]\[ x-8 = 3 \quad \text{or} \quad x-8 = -3 \][/tex]
Solving these gives:
[tex]\[ x = 11 \quad \text{or} \quad x = 5 \][/tex]
So, there are 2 solutions to this equation.
### Equation 2: [tex]\( 3-|x+4|=10 \)[/tex]
We start with:
[tex]\[ 3 - |x+4| = 10 \][/tex]
To isolate the absolute value term, subtract 3 from both sides:
[tex]\[ -|x+4| = 7 \][/tex]
Divide both sides by -1 to solve for the absolute value term:
[tex]\[ |x+4| = -7 \][/tex]
However, an absolute value cannot be a negative number. Therefore, there are no values of [tex]\(x\)[/tex] that satisfy this equation.
So, there are 0 solutions to this equation.
### Equation 3: [tex]\( \frac{1}{2}|x|+3=3 \)[/tex]
We start with:
[tex]\[ \frac{1}{2}|x| + 3 = 3 \][/tex]
Subtract 3 from both sides to isolate the term with the absolute value:
[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 equation is:
[tex]\[ x = 0 \][/tex]
So, there is 1 solution to this equation.
### Putting the Equations in Order
Now, let's arrange the equations in order from the least to the greatest number of solutions:
1. [tex]\( 3 - |x+4| = 10 \)[/tex] has 0 solutions.
2. [tex]\( \frac{1}{2}|x| + 3 = 3 \)[/tex] has 1 solution.
3. [tex]\( |x-8| - 4 = -1 \)[/tex] has 2 solutions.
Therefore, in order from least to greatest number of solutions:
[tex]\[ 3 - |x+4| = 10 \quad \rightarrow \quad \frac{1}{2}|x| + 3 = 3 \quad \rightarrow \quad |x-8| - 4 = -1 \][/tex]
