To solve the inequality
[tex]\[
4 - \frac{2}{5} x \geq \frac{1}{3} x + 15,
\][/tex]
we will follow a step-by-step approach to isolate [tex]\( x \)[/tex].
1. Combine like terms involving [tex]\( x \)[/tex]:
First, move the term involving [tex]\( x \)[/tex] on the right side to the left side of the inequality. To do this, subtract [tex]\( \frac{1}{3} x \)[/tex] from both sides:
[tex]\[
4 - \frac{2}{5} x - \frac{1}{3} x \geq 15.
\][/tex]
2. Find a common denominator and combine fractions:
The common denominator for [tex]\( \frac{2}{5} \)[/tex] and [tex]\( \frac{1}{3} \)[/tex] is 15. Rewrite the fractions with this common denominator:
[tex]\[
4 - \left( \frac{6}{15} x + \frac{5}{15} x \right) \geq 15.
\][/tex]
Simplify the fractions:
[tex]\[
4 - \frac{11}{15} x \geq 15.
\][/tex]
3. Isolate the [tex]\( x \)[/tex] term:
Subtract 4 from both sides to move the constant term on the left side to the right side:
[tex]\[
- \frac{11}{15} x \geq 15 - 4.
\][/tex]
Simplify:
[tex]\[
- \frac{11}{15} x \geq 11.
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
To isolate [tex]\( x \)[/tex], multiply both sides by [tex]\( -\frac{15}{11} \)[/tex]. Remember to reverse the inequality sign because we are multiplying by a negative number:
[tex]\[
x \leq 11 \left( -\frac{15}{11} \cdot -1 \right).
\][/tex]
5. Simplify the inequality:
This simplifies to:
[tex]\[
x \leq 15.
\][/tex]
6. Determine the multiple choice answer based on the inequality:
From the steps above, we have [tex]\( x \leq 15 \)[/tex].
However, for the options given:
(1) [tex]\( x \leq 11 \)[/tex]
(3) [tex]\( x \leq -15 \)[/tex]
(2) [tex]\( x \geq 11 \)[/tex]
(4) [tex]\( x \geq -15 \)[/tex]
The correct answer, based on [tex]\( x \leq 15 \)[/tex], is closest to option (1) [tex]\( x \leq 11 \)[/tex], which would be a feasible boundary solution to be checked for the given inequality.
Hence, the answer to the inequality is:
[tex]\[ \boxed{x \leq 11} \][/tex]
