To solve the equation
[tex]\[
\frac{3}{2} (x - 1) = \frac{4}{3} x,
\][/tex]
we will follow these steps:
1. Clear the fractions: The first thing to do is to eliminate the fractions by finding a common denominator. In this case, the common denominator for the fractions [tex]\( \frac{3}{2} \)[/tex] and [tex]\( \frac{4}{3} \)[/tex] is 6.
2. Multiply through by the common denominator: Multiply each term on both sides of the equation by 6 to clear the fractions:
[tex]\[
6 \cdot \frac{3}{2} (x - 1) = 6 \cdot \frac{4}{3} x.
\][/tex]
This simplifies to:
[tex]\[
3 \cdot 3 (x - 1) = 2 \cdot 4 x,
\][/tex]
or
[tex]\[
9 (x - 1) = 8 x.
\][/tex]
3. Distribute and simplify: Distribute the 9 on the left-hand side:
[tex]\[
9x - 9 = 8x.
\][/tex]
4. Move all terms involving [tex]\( x \)[/tex] to one side: To isolate [tex]\( x \)[/tex], we need to get all the [tex]\( x \)[/tex]-terms on one side and all the constant terms on the other side. Subtract [tex]\( 8x \)[/tex] from both sides:
[tex]\[
9x - 9 - 8x = 8x - 8x,
\][/tex]
which simplifies to:
[tex]\[
x - 9 = 0.
\][/tex]
5. Solve for [tex]\( x \)[/tex]: Isolate [tex]\( x \)[/tex] by adding 9 to both sides:
[tex]\[
x = 9.
\][/tex]
Thus, the solution to the equation
[tex]\[
\frac{3}{2} (x - 1) = \frac{4}{3} x
\][/tex]
is
[tex]\[
x = 9.
\][/tex]
