Certainly! Let's solve the equation [tex]\( | x - 5 + x | = 1 \)[/tex] step-by-step.
First, simplify the expression inside the absolute value:
[tex]\[ | x - 5 + x | = | 2x - 5 | = 1 \][/tex]
Next, recall that the absolute value equation [tex]\( |A| = B \)[/tex] can be written as two separate equations:
[tex]\[ A = B \quad \text{or} \quad A = -B \][/tex]
Here, our [tex]\( A \)[/tex] is [tex]\( 2x - 5 \)[/tex] and [tex]\( B \)[/tex] is 1. So, we split our equation into two cases:
1. [tex]\( 2x - 5 = 1 \)[/tex]
2. [tex]\( 2x - 5 = -1 \)[/tex]
### Solving the First Case: [tex]\( 2x - 5 = 1 \)[/tex]
1. Add 5 to both sides:
[tex]\[ 2x - 5 + 5 = 1 + 5 \][/tex]
[tex]\[ 2x = 6 \][/tex]
2. Divide both sides by 2:
[tex]\[ \frac{2x}{2} = \frac{6}{2} \][/tex]
[tex]\[ x = 3 \][/tex]
So, one solution is [tex]\( x = 3 \)[/tex].
### Solving the Second Case: [tex]\( 2x - 5 = -1 \)[/tex]
1. Add 5 to both sides:
[tex]\[ 2x - 5 + 5 = -1 + 5 \][/tex]
[tex]\[ 2x = 4 \][/tex]
2. Divide both sides by 2:
[tex]\[ \frac{2x}{2} = \frac{4}{2} \][/tex]
[tex]\[ x = 2 \][/tex]
So, the other solution is [tex]\( x = 2 \)[/tex].
### Conclusion
The solutions to the equation [tex]\( |2x - 5| = 1 \)[/tex] are:
[tex]\[ x = 2 \quad \text{and} \quad x = 3 \][/tex]
