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Answer
La solución es \( x = 0 \) y \( x = \frac{2}{3} \).
Solution
a) \( |x-1|=2x-1 \)
Para resolver esta desigualdad, primero consideramos que el valor absoluto de una expresión es igual a una cantidad positiva si y solo si la expresión puede ser igual a esa cantidad positiva o a su negativo. Por lo tanto, podemos descomponer la desigualdad en dos casos:
1. \( x - 1 = 2x - 1 \)
2. \( x - 1 = -(2x - 1) \)
Resolvamos cada caso:
1. \( x - 1 = 2x - 1 \)
\[
x - 2x = -1 + 1 \\
-x = 0 \\
x = 0
\]
2. \( x - 1 = -(2x - 1) \)
\[
x - 1 = -2x + 1 \\
x + 2x = 1 + 1 \\
3x = 2 \\
x = \frac{2}{3}
\]
Ahora, verificamos estos valores en la desigualdad original:
Para \( x = 0 \):
\[
|x - 1| = |0 - 1| = |-1| = 1 \\
2x - 1 = 2(0) - 1 = -1
\]
Por lo tanto, \( |x - 1| = 2x - 1 \) es verdadero para \( x = 0 \).
Para \( x = \frac{2}{3} \):
\[
|x - 1| = \left|\frac{2}{3} - 1\right| = \left|-\frac{1}{3}\right| = \frac{1}{3} \\
2x - 1 = 2\left(\frac{2}{3}\right) - 1 = \frac{4}{3} - 1 = \frac{1}{3}
\]
Por lo tanto, \( |x - 1| = 2x - 1 \) es verdadero para \( x = \frac{2}{3} \).
Ambos valores, \( x = 0 \) y \( x = \frac{2}{3} \), satisfacen la desigualdad original. Por lo tanto, la solución es \( x = 0 \) y \( x = \frac{2}{3} \).
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Simplify this solution Bonus Knowledge
To solve the equation \( |x-1|=2x-1 \), we can break it down into two cases based on the definition of absolute value. **Case 1:** When \( x-1 \geq 0 \) (which simplifies to \( x \geq 1 \)), the equation becomes \( x-1=2x-1 \). Solving this gives \( x=0 \), which doesn't satisfy \( x \geq 1 \). **Case 2:** When \( x-1 < 0 \) (or \( x < 1 \)), the equation simplifies to \( -(x-1) = 2x - 1 \), leading to \( -x + 1 = 2x - 1 \). Solving this gives \( 3x = 2 \), or \( x = \frac{2}{3} \), which satisfies \( x < 1 \) and is thus a valid solution. So, the solution to \( |x-1|=2x-1 \) is \( x = \frac{2}{3} \).
