Let's solve the inequality step-by-step:
[tex]\[
\frac{x}{3} + \frac{x}{2} > 5 - \frac{x}{6}
\][/tex]
1. Combine the fractions on the left-hand side:
First, we need a common denominator for the fractions. The common denominator between 3, 2, and 6 is 6.
Rewrite each fraction with the common denominator:
[tex]\[
\frac{x}{3} = \frac{2x}{6}, \quad \frac{x}{2} = \frac{3x}{6}, \quad \frac{x}{6} \text{ remains the same}
\][/tex]
So the inequality becomes:
[tex]\[
\frac{2x}{6} + \frac{3x}{6} > 5 - \frac{x}{6}
\][/tex]
Combine the fractions on the left-hand side:
[tex]\[
\frac{2x + 3x}{6} > 5 - \frac{x}{6}
\][/tex]
[tex]\[
\frac{5x}{6} > 5 - \frac{x}{6}
\][/tex]
2. Eliminate the fraction by multiplying through by 6:
Multiply all terms in the inequality by 6 to clear the denominators:
[tex]\[
6 \cdot \frac{5x}{6} > 6 \cdot 5 - 6 \cdot \frac{x}{6}
\][/tex]
[tex]\[
5x > 30 - x
\][/tex]
3. Isolate the variable x:
Add [tex]\( x \)[/tex] to both sides of the inequality to get all the [tex]\( x \)[/tex]-terms on one side:
[tex]\[
5x + x > 30
\][/tex]
[tex]\[
6x > 30
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
Divide both sides by 6:
[tex]\[
x > 5
\][/tex]
Therefore, the solution to the inequality is:
[tex]\[
x > 5
\][/tex]
In interval notation, this is expressed as:
[tex]\[
(5, \infty)
\][/tex]
So, the answer to the inequality [tex]\(\frac{x}{3} + \frac{x}{2} > 5 - \frac{x}{6}\)[/tex] is [tex]\(x > 5\)[/tex].
