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The inequality [tex]6 - \frac{2}{3}x < x - 9[/tex] is equivalent to:

A. [tex]x > 9[/tex]

B. [tex]x < 9[/tex]

C. [tex]x > -\frac{3}{5}[/tex]

D. [tex]x < -\frac{3}{5}[/tex]

Answer :

We are given the inequality:
$$
6 - \frac{2}{3}x < x - 9.
$$

Follow these steps:

1. Multiply every term by 3 to eliminate the fraction:
$$
3 \cdot 6 - 3 \cdot \frac{2}{3}x < 3x - 3 \cdot 9,
$$
which simplifies to:
$$
18 - 2x < 3x - 27.
$$

2. Next, add $2x$ to both sides to bring the variable terms together:
$$
18 < 3x + 2x - 27,
$$
which simplifies to:
$$
18 < 5x - 27.
$$

3. Now, add 27 to both sides to isolate the term with $x$:
$$
18 + 27 < 5x,
$$
so:
$$
45 < 5x.
$$

4. Finally, divide both sides by 5:
$$
\frac{45}{5} < x,
$$
hence:
$$
9 < x,
$$
which can be written as:
$$
x > 9.
$$

Thus, the inequality $6-\frac{2}{3} x < x-9$ is equivalent to:
$$
x > 9.
$$

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