Answer :
Let's solve the inequality [tex]\(31 \leq 3k + 1 < 151\)[/tex] step by step:
1. Break down the compound inequality into two separate inequalities:
- [tex]\(31 \leq 3k + 1\)[/tex]
- [tex]\(3k + 1 < 151\)[/tex]
2. Solve the first inequality [tex]\(31 \leq 3k + 1\)[/tex]:
- Subtract 1 from both sides:
[tex]\[
31 - 1 \leq 3k
\][/tex]
[tex]\[
30 \leq 3k
\][/tex]
- Divide both sides by 3:
[tex]\[
10 \leq k
\][/tex]
- So the solution for the first inequality is [tex]\(k \geq 10\)[/tex].
3. Solve the second inequality [tex]\(3k + 1 < 151\)[/tex]:
- Subtract 1 from both sides:
[tex]\[
3k < 150
\][/tex]
- Divide both sides by 3:
[tex]\[
k < 50
\][/tex]
- So the solution for the second inequality is [tex]\(k < 50\)[/tex].
4. Combine the solutions:
- The solutions from both inequalities are [tex]\(k \geq 10\)[/tex] and [tex]\(k < 50\)[/tex].
- Combining these gives a solution of:
[tex]\[
10 \leq k < 50
\][/tex]
So, the solution for the inequality [tex]\(31 \leq 3k + 1 < 151\)[/tex] is [tex]\(10 \leq k < 50\)[/tex]. This means [tex]\(k\)[/tex] can be any value from 10 to just less than 50.
1. Break down the compound inequality into two separate inequalities:
- [tex]\(31 \leq 3k + 1\)[/tex]
- [tex]\(3k + 1 < 151\)[/tex]
2. Solve the first inequality [tex]\(31 \leq 3k + 1\)[/tex]:
- Subtract 1 from both sides:
[tex]\[
31 - 1 \leq 3k
\][/tex]
[tex]\[
30 \leq 3k
\][/tex]
- Divide both sides by 3:
[tex]\[
10 \leq k
\][/tex]
- So the solution for the first inequality is [tex]\(k \geq 10\)[/tex].
3. Solve the second inequality [tex]\(3k + 1 < 151\)[/tex]:
- Subtract 1 from both sides:
[tex]\[
3k < 150
\][/tex]
- Divide both sides by 3:
[tex]\[
k < 50
\][/tex]
- So the solution for the second inequality is [tex]\(k < 50\)[/tex].
4. Combine the solutions:
- The solutions from both inequalities are [tex]\(k \geq 10\)[/tex] and [tex]\(k < 50\)[/tex].
- Combining these gives a solution of:
[tex]\[
10 \leq k < 50
\][/tex]
So, the solution for the inequality [tex]\(31 \leq 3k + 1 < 151\)[/tex] is [tex]\(10 \leq k < 50\)[/tex]. This means [tex]\(k\)[/tex] can be any value from 10 to just less than 50.
