To solve the inequality [tex]\(12 < \frac{4x + 12}{3} < 16\)[/tex], follow these steps carefully:
1. Eliminate the fraction: We start by multiplying all parts of the inequality by 3 to get rid of the fraction. This results in:
[tex]\[
12 \times 3 < \left(\frac{4x + 12}{3}\right) \times 3 < 16 \times 3
\][/tex]
Simplifying, we have:
[tex]\[
36 < 4x + 12 < 48
\][/tex]
2. Isolate the term containing [tex]\(x\)[/tex]: Next, we need to isolate [tex]\(4x\)[/tex] in the middle. To do this, subtract 12 from all parts of the inequality:
[tex]\[
36 - 12 < 4x + 12 - 12 < 48 - 12
\][/tex]
Simplifying, we get:
[tex]\[
24 < 4x < 36
\][/tex]
3. Solve for [tex]\(x\)[/tex]: Now, divide all parts of the inequality by 4 to solve for [tex]\(x\)[/tex]:
[tex]\[
\frac{24}{4} < \frac{4x}{4} < \frac{36}{4}
\][/tex]
Simplifying, we have:
[tex]\[
6 < x < 9
\][/tex]
Therefore, the solution to the inequality is:
[tex]\[
6 < x < 9
\][/tex]
