To solve the inequality [tex]\(\frac{1}{3}\left(\frac{2}{5} x + 1\right) \geq \frac{1}{5}(x + 4)\)[/tex], we need to follow several steps to manipulate the expression into a more workable form and solve for [tex]\(x\)[/tex]. Here is the step-by-step process:
1. Distribute and Combine Like Terms:
First, we distribute the fractions within the parentheses.
[tex]\[
\frac{1}{3}\left(\frac{2}{5} x + 1\right) = \frac{1}{3} \cdot \frac{2}{5} x + \frac{1}{3} \cdot 1 = \frac{2}{15} x + \frac{1}{3}
\][/tex]
Similarly, for the right side:
[tex]\[
\frac{1}{5}(x + 4) = \frac{1}{5} x + \frac{4}{5}
\][/tex]
2. Set Up the Inequality:
Now the inequality becomes:
[tex]\[
\frac{2}{15} x + \frac{1}{3} \geq \frac{1}{5} x + \frac{4}{5}
\][/tex]
3. Eliminate Fractions by Finding a Common Denominator:
To simplify calculations, eliminate the fractions by multiplying everything by a common denominator. In this case, the smallest common multiple of 15, 3, and 5 is 15.
[tex]\[
15 \left(\frac{2}{15} x + \frac{1}{3}\right) \geq 15 \left(\frac{1}{5} x + \frac{4}{5}\right)
\][/tex]
Distributing the 15:
[tex]\[
15 \cdot \frac{2}{15} x + 15 \cdot \frac{1}{3} \geq 15 \cdot \frac{1}{5} x + 15 \cdot \frac{4}{5}
\][/tex]
Simplifying further:
[tex]\[
2x + 5 \geq 3x + 12
\][/tex]
4. Isolate [tex]\(x\)[/tex] on One Side:
To isolate [tex]\(x\)[/tex], subtract [tex]\(2x\)[/tex] from both sides:
[tex]\[
5 \geq x + 12
\][/tex]
Then subtract 12 from both sides:
[tex]\[
5 - 12 \geq x
\][/tex]
Which simplifies to:
[tex]\[
-7 \geq x
\][/tex]
5. Rewrite the Solution:
This inequality can be rewritten as:
[tex]\[
x \leq -7
\][/tex]
Thus, the solution to the inequality [tex]\(\frac{1}{3}\left(\frac{2}{5} x + 1\right) \geq \frac{1}{5}(x + 4)\)[/tex] is:
[tex]\[
x \leq -7
\][/tex]
In interval notation, this solution set is [tex]\((- \infty, -7]\)[/tex].
So the complete solution of the inequality is:
[tex]\[ x \leq -7 \][/tex]
