Sure, let's solve the given inequalities step-by-step to determine the ranges for \( t \) such that the radicand (expression inside the square root) remains non-negative.
### Inequality 1: \( 3t - 9 \geq 0 \)
1. Start with the inequality:
[tex]\[
3t - 9 \geq 0
\][/tex]
2. Add 9 to both sides to isolate the term with \( t \):
[tex]\[
3t \geq 9
\][/tex]
3. Divide both sides by 3:
[tex]\[
t \geq 3
\][/tex]
So, the solution for the first inequality is:
[tex]\[
t \geq 3
\][/tex]
### Inequality 2: \( 3t + 9 \geq 0 \)
1. Start with the inequality:
[tex]\[
3t + 9 \geq 0
\][/tex]
2. Subtract 9 from both sides:
[tex]\[
3t \geq -9
\][/tex]
3. Divide both sides by 3:
[tex]\[
t \geq -3
\][/tex]
So, the solution for the second inequality is:
[tex]\[
t \geq -3
\][/tex]
### Inequality 3: \( 3t \geq 0 \)
1. Start with the inequality:
[tex]\[
3t \geq 0
\][/tex]
2. Divide both sides by 3:
[tex]\[
t \geq 0
\][/tex]
So, the solution for the third inequality is:
[tex]\[
t \geq 0
\][/tex]
### Inequality 4: \( t \geq 0 \)
This is already solved, and the solution is:
[tex]\[
t \geq 0
\][/tex]
### Conclusion
Combining all these inequalities, we get the ranges that satisfy each inequality:
1. \( t \geq 3 \).
2. \( t \geq -3 \).
3. \( t \geq 0 \).
4. \( t \geq 0 \).
Since the tightest constraint among these inequalities is \( t \geq 3 \), the combined solution that satisfies all the inequalities is:
[tex]\[
t \geq 3
\][/tex]
Conclusively, the radicand will be non-negative if [tex]\( t \geq 3 \)[/tex].
