To solve the inequality [tex]\( 4 - \frac{2}{5}x \geq \frac{1}{3}x + 15 \)[/tex], follow these steps:
1. Move all terms involving [tex]\( x \)[/tex] to one side of the inequality:
[tex]\[
4 - 15 \geq \frac{1}{3}x + \frac{2}{5}x
\][/tex]
2. Simplify the constants on the left-hand side:
[tex]\[
4 - 15 = -11
\][/tex]
So we have:
[tex]\[
-11 \geq \frac{1}{3}x + \frac{2}{5}x
\][/tex]
3. Combine the [tex]\( x \)[/tex]-terms on the right:
[tex]\[
\frac{1}{3}x + \frac{2}{5}x
\][/tex]
To combine these fractions, find a common denominator. The least common multiple of 3 and 5 is 15:
[tex]\[
\frac{1}{3}x = \frac{5}{15}x \quad \text{and} \quad \frac{2}{5}x = \frac{6}{15}x
\][/tex]
So:
[tex]\[
\frac{5}{15}x + \frac{6}{15}x = \frac{11}{15}x
\][/tex]
This gives us:
[tex]\[
-11 \geq \frac{11}{15}x
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
To isolate [tex]\( x \)[/tex], divide both sides by [tex]\( \frac{11}{15} \)[/tex]:
[tex]\[
x = \frac{-11}{\frac{11}{15}}
\][/tex]
5. Simplify the division:
[tex]\[
x = -11 \times \frac{15}{11}
\][/tex]
Simplifying further:
[tex]\[
x = -15
\][/tex]
Therefore, the solution to the inequality [tex]\( 4 - \frac{2}{5}x \geq \frac{1}{3}x + 15 \)[/tex] is [tex]\( x \leq -15 \)[/tex].
The correct answer is:
- (3) [tex]\( x \leq -15 \)[/tex]
