Question

how many solutions exist for the given equation?
|3x - 2| = 2 - x
infinitely many
zero
two
one

Explanation:

Step1: Analyze the absolute value equation

We have the equation \(|x - 2|=-2 - x\). Recall that the absolute value \(|a|\) is always non - negative, i.e., \(|a|\geq0\) for all real numbers \(a\). So, \(|x - 2|\geq0\) for all real \(x\).

Step2: Analyze the right - hand side

Now, consider the right - hand side of the equation \(y=-2 - x\). We want to find when \(-2 - x\geq0\). Solving the inequality \(-2 - x\geq0\), we add \(x\) to both sides: \(-2\geq x\), or \(x\leq - 2\).

When \(x\leq - 2\), \(x - 2\) is negative (because if \(x\leq - 2\), then \(x-2\leq-2 - 2=-4<0\)). So, \(|x - 2|=-(x - 2)=-x + 2\) when \(x\leq - 2\).

Substitute \(|x - 2|=-x + 2\) into the original equation:
\(-x + 2=-2 - x\)

Step3: Solve the resulting equation

Add \(x\) to both sides of the equation \(-x + 2=-2 - x\). We get \(2=-2\), which is a contradiction.

This means that there are no values of \(x\) that satisfy the original equation.

Answer:

zero