Answer :
To solve the inequality [tex]\(6 - \frac{2}{3}x < x - 9\)[/tex] and find the equivalent expression, we can follow these steps:
1. Start with the original inequality:
[tex]\[
6 - \frac{2}{3}x < x - 9
\][/tex]
2. Eliminate fractions by multiplying every term by 3, the least common multiple of the denominators:
[tex]\[
3(6) - 2x < 3x - 3(9)
\][/tex]
Simplifying, we get:
[tex]\[
18 - 2x < 3x - 27
\][/tex]
3. Rearrange the inequality to isolate terms involving [tex]\(x\)[/tex] on one side. Add [tex]\(2x\)[/tex] to both sides:
[tex]\[
18 < 3x + 2x - 27
\][/tex]
Simplifying this gives:
[tex]\[
18 < 5x - 27
\][/tex]
4. Add 27 to both sides to further isolate the [tex]\(x\)[/tex] term:
[tex]\[
18 + 27 < 5x
\][/tex]
Simplifying, we get:
[tex]\[
45 < 5x
\][/tex]
5. Divide both sides by 5 to solve for [tex]\(x\)[/tex]:
[tex]\[
\frac{45}{5} < x
\][/tex]
Simplifying, we find:
[tex]\[
9 < x
\][/tex]
6. Rewriting the solution, we have:
[tex]\[
x > 9
\][/tex]
Therefore, the inequality [tex]\(6 - \frac{2}{3}x < x - 9\)[/tex] is equivalent to [tex]\(x > 9\)[/tex]. So, the correct choice is [tex]\(x > 9\)[/tex].
1. Start with the original inequality:
[tex]\[
6 - \frac{2}{3}x < x - 9
\][/tex]
2. Eliminate fractions by multiplying every term by 3, the least common multiple of the denominators:
[tex]\[
3(6) - 2x < 3x - 3(9)
\][/tex]
Simplifying, we get:
[tex]\[
18 - 2x < 3x - 27
\][/tex]
3. Rearrange the inequality to isolate terms involving [tex]\(x\)[/tex] on one side. Add [tex]\(2x\)[/tex] to both sides:
[tex]\[
18 < 3x + 2x - 27
\][/tex]
Simplifying this gives:
[tex]\[
18 < 5x - 27
\][/tex]
4. Add 27 to both sides to further isolate the [tex]\(x\)[/tex] term:
[tex]\[
18 + 27 < 5x
\][/tex]
Simplifying, we get:
[tex]\[
45 < 5x
\][/tex]
5. Divide both sides by 5 to solve for [tex]\(x\)[/tex]:
[tex]\[
\frac{45}{5} < x
\][/tex]
Simplifying, we find:
[tex]\[
9 < x
\][/tex]
6. Rewriting the solution, we have:
[tex]\[
x > 9
\][/tex]
Therefore, the inequality [tex]\(6 - \frac{2}{3}x < x - 9\)[/tex] is equivalent to [tex]\(x > 9\)[/tex]. So, the correct choice is [tex]\(x > 9\)[/tex].
