Let's solve the equation [tex]\( |x| + |4x - 4| + x = 14 \)[/tex] step-by-step by considering different cases for the absolute value functions.
### Case 1: [tex]\( x \ge 1 \)[/tex]
- In this case, [tex]\( |x| = x \)[/tex] because [tex]\( x \)[/tex] is non-negative.
- Also, [tex]\( |4x - 4| = 4x - 4 \)[/tex] because [tex]\( 4x - 4 \ge 0 \)[/tex].
So the equation simplifies to:
[tex]\[ x + (4x - 4) + x = 14 \][/tex]
Combine like terms:
[tex]\[ x + 4x - 4 + x = 14 \][/tex]
[tex]\[ 6x - 4 = 14 \][/tex]
Adding 4 to both sides gives:
[tex]\[ 6x = 18 \][/tex]
Divide both sides by 6:
[tex]\[ x = 3 \][/tex]
Check that [tex]\( x = 3 \)[/tex] satisfies the condition [tex]\( x \ge 1 \)[/tex]. It does, so [tex]\( x = 3 \)[/tex] is a valid solution for this case.
### Case 2: [tex]\( 0 \le x < 1 \)[/tex]
- Here, [tex]\( |x| = x \)[/tex] because [tex]\( x \)[/tex] is non-negative.
- However, [tex]\( |4x - 4| = 4 - 4x \)[/tex] because [tex]\( 4x - 4 \)[/tex] is negative in this range ([tex]\( x < 1 \)[/tex]).
So the equation simplifies to:
[tex]\[ x + (4 - 4x) + x = 14 \][/tex]
Combine like terms:
[tex]\[ x + 4 - 4x + x = 14 \][/tex]
[tex]\[ 2x + 4 - 4x = 14 \][/tex]
[tex]\[ -2x + 4 = 14 \][/tex]
Subtract 4 from both sides:
[tex]\[ -2x = 10 \][/tex]
Divide both sides by -2:
[tex]\[ x = -5 \][/tex]
Check the condition [tex]\( 0 \le x < 1 \)[/tex] for [tex]\( x = -5 \)[/tex]. It does not satisfy the condition, so [tex]\( x = -5 \)[/tex] is not a valid solution for this case.
### Case 3: [tex]\( x < 0 \)[/tex]
- In this case, [tex]\( |x| = -x \)[/tex] because [tex]\( x \)[/tex] is negative.
- Also, [tex]\( |4x - 4| = 4 - 4x \)[/tex] because [tex]\( 4x - 4 \)[/tex] is negative if [tex]\( x \)[/tex] is negative.
So the equation simplifies to:
[tex]\[ -x + (4 - 4x) + x = 14 \][/tex]
Combine like terms:
[tex]\[ -x + 4 - 4x + x = 14 \][/tex]
[tex]\[ 4 - 4x = 14 \][/tex]
Subtract 4 from both sides:
[tex]\[ -4x = 10 \][/tex]
Divide both sides by -4:
[tex]\[ x = -\frac{10}{4} \][/tex]
[tex]\[ x = -\frac{5}{2} \][/tex]
Check that [tex]\( x = -\frac{5}{2} \)[/tex] satisfies the condition [tex]\( x < 0 \)[/tex]. It does, so [tex]\( x = -\frac{5}{2} \)[/tex] is a valid solution for this case.
### Conclusion
The solutions to the equation [tex]\( |x| + |4x - 4| + x = 14 \)[/tex] are:
[tex]\[ x = 3 \][/tex]
[tex]\[ x = -\frac{5}{2} \][/tex]
