Answer :
To solve the inequality [tex]\(6 - \frac{2}{3}x < x - 9\)[/tex], we follow these steps:
1. Eliminate Fractions:
Start by clearing the fraction. You can do this by multiplying every term in the inequality by 3 to eliminate the denominator.
[tex]\[
3 \cdot 6 - 3 \cdot \frac{2}{3} x < 3(x - 9)
\][/tex]
Simplifying gives:
[tex]\[
18 - 2x < 3x - 27
\][/tex]
2. Rearrange Terms:
Next, we'll move all the terms involving [tex]\(x\)[/tex] to one side of the inequality and constants to the other side. To do this, add [tex]\(2x\)[/tex] to both sides and add 27 to both sides:
[tex]\[
18 + 27 < 3x + 2x
\][/tex]
Simplifying yields:
[tex]\[
45 < 5x
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Now, divide both sides of the inequality by 5 to solve for [tex]\(x\)[/tex]:
[tex]\[
\frac{45}{5} < x
\][/tex]
Simplifying gives:
[tex]\[
9 < x
\][/tex]
Which can also be written as:
[tex]\[
x > 9
\][/tex]
Therefore, the inequality is equivalent to [tex]\(x > 9\)[/tex].
1. Eliminate Fractions:
Start by clearing the fraction. You can do this by multiplying every term in the inequality by 3 to eliminate the denominator.
[tex]\[
3 \cdot 6 - 3 \cdot \frac{2}{3} x < 3(x - 9)
\][/tex]
Simplifying gives:
[tex]\[
18 - 2x < 3x - 27
\][/tex]
2. Rearrange Terms:
Next, we'll move all the terms involving [tex]\(x\)[/tex] to one side of the inequality and constants to the other side. To do this, add [tex]\(2x\)[/tex] to both sides and add 27 to both sides:
[tex]\[
18 + 27 < 3x + 2x
\][/tex]
Simplifying yields:
[tex]\[
45 < 5x
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Now, divide both sides of the inequality by 5 to solve for [tex]\(x\)[/tex]:
[tex]\[
\frac{45}{5} < x
\][/tex]
Simplifying gives:
[tex]\[
9 < x
\][/tex]
Which can also be written as:
[tex]\[
x > 9
\][/tex]
Therefore, the inequality is equivalent to [tex]\(x > 9\)[/tex].
