Certainly! Let's solve the equation [tex]\( |x| + |4x - 4| + x = 14 \)[/tex] step-by-step.
Given the absolute value expressions, we need to consider different cases based on the values of [tex]\(x\)[/tex]:
1. [tex]\(x \geq 1\)[/tex]
2. [tex]\(0 \leq x < 1\)[/tex]
3. [tex]\(x < 0\)[/tex]
### Case 1: [tex]\( x \geq 1 \)[/tex]
In this range:
- [tex]\( |x| = x \)[/tex]
- [tex]\( |4x - 4| = 4x - 4 \)[/tex]
Therefore, the equation [tex]\( |x| + |4x - 4| + x = 14 \)[/tex] simplifies to:
[tex]\[ x + (4x - 4) + x = 14 \][/tex]
[tex]\[ 6x - 4 = 14 \][/tex]
[tex]\[ 6x = 18 \][/tex]
[tex]\[ x = 3 \][/tex]
Since [tex]\( x = 3 \)[/tex] lies within [tex]\( x \geq 1 \)[/tex], it is a valid solution for this case.
### Case 2: [tex]\( 0 \leq x < 1 \)[/tex]
In this range:
- [tex]\( |x| = x \)[/tex]
- [tex]\( |4x - 4| = 4 - 4x \)[/tex] (since [tex]\(4x - 4\)[/tex] is negative in this range)
Therefore, the equation [tex]\( |x| + |4x - 4| + x = 14 \)[/tex] simplifies to:
[tex]\[ x + (4 - 4x) + x = 14 \][/tex]
[tex]\[ x + 4 - 4x + x = 14 \][/tex]
[tex]\[ 2x + 4 - 4x = 14 \][/tex]
[tex]\[ -2x + 4 = 14 \][/tex]
[tex]\[ -2x = 10 \][/tex]
[tex]\[ x = -5 \][/tex]
Since [tex]\( x = -5 \)[/tex] does not lie within [tex]\( 0 \leq x < 1 \)[/tex], it is not a valid solution for this case.
### Case 3: [tex]\( x < 0 \)[/tex]
In this range:
- [tex]\( |x| = -x \)[/tex]
- [tex]\( |4x - 4| = 4 - 4x \)[/tex] (since [tex]\(4x - 4\)[/tex] is negative in this range)
Therefore, the equation [tex]\( |x| + |4x - 4| + x = 14 \)[/tex] simplifies to:
[tex]\[ -x + (4 - 4x) + x = 14 \][/tex]
[tex]\[ -x + 4 - 4x + x = 14 \][/tex]
[tex]\[ 4 - 4x = 14 \][/tex]
[tex]\[ -4x = 10 \][/tex]
[tex]\[ x = - \frac{5}{2} \][/tex]
Since [tex]\( x = - \frac{5}{2} \)[/tex] lies within [tex]\( x < 0 \)[/tex], it is a valid solution for this case.
### Summary of Solutions:
After evaluating all cases, the solutions to the equation [tex]\( |x| + |4x - 4| + x = 14 \)[/tex] are:
[tex]\[ x = 3 \][/tex]
and
[tex]\[ x = -\frac{5}{2} \][/tex]
Hence, the solutions are [tex]\(\boxed{3 \text{ and } -\frac{5}{2}}\)[/tex].
