Let's solve the equation [tex]\( \frac{2}{3} x - \frac{1}{2} x = -\frac{1}{6} x + 2 \)[/tex] step by step to determine the number of days, [tex]\( x \)[/tex], that Michael has to wait to receive his car.
1. Combine like terms on the left-hand side:
[tex]\[
\frac{2}{3} x - \frac{1}{2} x
\][/tex]
To combine these fractions, we need a common denominator. The common denominator for 3 and 2 is 6. Rewriting the fractions with this common denominator:
[tex]\[
\frac{2}{3} x = \frac{4}{6} x
\][/tex]
[tex]\[
\frac{1}{2} x = \frac{3}{6} x
\][/tex]
Now, substitute these back into the equation:
[tex]\[
\frac{4}{6} x - \frac{3}{6} x = -\frac{1}{6} x + 2
\][/tex]
2. Simplify the left-hand side:
[tex]\[
\frac{4}{6} x - \frac{3}{6} x = \frac{1}{6} x
\][/tex]
So the equation now looks like:
[tex]\[
\frac{1}{6} x = -\frac{1}{6} x + 2
\][/tex]
3. Move all terms involving [tex]\( x \)[/tex] to one side of the equation:
Add [tex]\(\frac{1}{6} x\)[/tex] to both sides to isolate the [tex]\( x \)[/tex] terms:
[tex]\[
\frac{1}{6} x + \frac{1}{6} x = 2
\][/tex]
Simplifying, we get:
[tex]\[
\frac{2}{6} x = 2
\][/tex]
Which simplifies further to:
[tex]\[
\frac{1}{3} x = 2
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
To isolate [tex]\( x \)[/tex], multiply both sides of the equation by 3:
[tex]\[
x = 2 \times 3
\][/tex]
[tex]\[
x = 6
\][/tex]
Thus, Michael will have to wait [tex]\( 6 \)[/tex] days to receive his car.
Answer:
[tex]\[
\boxed{6} \text{ Days}
\][/tex]
