To solve the inequality [tex]\(12 \leq 3y + 15 \leq 24\)[/tex], we need to break it down into two separate inequalities and solve each one individually.
1. Solve the left inequality [tex]\(12 \leq 3y + 15\)[/tex]:
Start by isolating [tex]\( y \)[/tex] on one side of the inequality.
[tex]\[
12 \leq 3y + 15
\][/tex]
Subtract 15 from both sides to get [tex]\( 3y \)[/tex] by itself:
[tex]\[
12 - 15 \leq 3y
\][/tex]
Simplify the left side:
[tex]\[
-3 \leq 3y
\][/tex]
Divide both sides by 3 to solve for [tex]\( y \)[/tex]:
[tex]\[
-1 \leq y
\][/tex]
2. Solve the right inequality [tex]\(3y + 15 \leq 24\)[/tex]:
Start by isolating [tex]\( y \)[/tex] on one side of the inequality.
[tex]\[
3y + 15 \leq 24
\][/tex]
Subtract 15 from both sides to get [tex]\( 3y \)[/tex] by itself:
[tex]\[
3y \leq 24 - 15
\][/tex]
Simplify the right side:
[tex]\[
3y \leq 9
\][/tex]
Divide both sides by 3 to solve for [tex]\( y \)[/tex]:
[tex]\[
y \leq 3
\][/tex]
3. Combine the inequalities:
Now we combine the results of the two inequalities:
[tex]\[
-1 \leq y \leq 3
\][/tex]
This means that [tex]\( y \)[/tex] must be greater than or equal to [tex]\(-1\)[/tex] and less than or equal to [tex]\(3\)[/tex].
4. Write the solution set in interval notation:
In interval notation, the solution is written as:
[tex]\[
[-1, 3]
\][/tex]
Hence, the solution to the inequality [tex]\( 12 \leq 3y + 15 \leq 24 \)[/tex] is [tex]\( y \in [-1, 3] \)[/tex].
