High School

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

A. [tex]x \ \textgreater \ 9[/tex]
B. [tex]x \ \textless \ 9[/tex]

Answer :

To solve the inequality [tex]\(6 - \frac{2}{3}x < x - 9\)[/tex], follow these steps:

1. Get all terms involving [tex]\(x\)[/tex] on one side:
Start by subtracting [tex]\(x\)[/tex] from both sides of the inequality to get:
[tex]\[
6 - \frac{2}{3}x - x < -9
\][/tex]

2. Combine like terms involving [tex]\(x\)[/tex]:
Combine [tex]\(-\frac{2}{3}x\)[/tex] and [tex]\(-x\)[/tex]. To do this, consider [tex]\(x\)[/tex] as [tex]\(\frac{3}{3}x\)[/tex]:
[tex]\[
-\frac{2}{3}x - \frac{3}{3}x = -\frac{5}{3}x
\][/tex]
So the inequality becomes:
[tex]\[
6 - \frac{5}{3}x < -9
\][/tex]

3. Isolate the term with [tex]\(x\)[/tex]:
Subtract 6 from both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[
-\frac{5}{3}x < -9 - 6
\][/tex]
Simplifying the right side gives:
[tex]\[
-\frac{5}{3}x < -15
\][/tex]

4. Solve for [tex]\(x\)[/tex]:
To solve for [tex]\(x\)[/tex], divide both sides by [tex]\(-\frac{5}{3}\)[/tex]. Remember, dividing or multiplying an inequality by a negative number reverses the inequality. So you have:
[tex]\[
x > \frac{-15}{-\frac{5}{3}}
\][/tex]
Simplifying gives:
[tex]\[
x > 9
\][/tex]

Thus, the solution to the inequality is [tex]\(x > 9\)[/tex].

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