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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 start with the inequality
[tex]$$
6 - \frac{2}{3}x < x - 9.
$$[/tex]

Step 1. Subtract 6 from both sides.
Subtracting 6 gives:
[tex]$$
6 - \frac{2}{3}x - 6 < x - 9 - 6,
$$[/tex]
which simplifies to:
[tex]$$
-\frac{2}{3}x < x - 15.
$$[/tex]

Step 2. Subtract [tex]$x$[/tex] from both sides.
Subtracting [tex]$x$[/tex] from each side yields:
[tex]$$
-\frac{2}{3}x - x < -15.
$$[/tex]
We can combine the [tex]$x$[/tex] terms on the left:
[tex]$$
-\frac{2}{3}x - \frac{3}{3}x = -\frac{5}{3}x.
$$[/tex]
So the inequality becomes:
[tex]$$
-\frac{5}{3}x < -15.
$$[/tex]

Step 3. Multiply both sides by [tex]$-\frac{3}{5}$[/tex].
When multiplying or dividing by a negative number, we must reverse the inequality sign. Thus,
[tex]$$
x > (-15) \cdot \left(-\frac{3}{5}\right).
$$[/tex]
Calculating the right side:
[tex]$$
(-15) \cdot \left(-\frac{3}{5}\right) = \frac{45}{5} = 9.
$$[/tex]
This leads us to the final inequality:
[tex]$$
x > 9.
$$[/tex]

Thus, the original inequality is equivalent to
[tex]$$
x > 9.
$$[/tex]

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