To set up an inequality to ensure that the radicand (the expression inside a square root) cannot be negative, we need to analyze the given inequalities step by step.
1. First Inequality:
[tex]\[
3t - 9 \geq 0
\][/tex]
This means that [tex]\(3t\)[/tex] must be at least 9. We solve it as follows:
[tex]\[
3t - 9 \geq 0 \implies 3t \geq 9 \implies t \geq 3
\][/tex]
2. Second Inequality:
[tex]\[
3t + 9 \geq 0
\][/tex]
Since we want the radicand to be non-negative, we solve this inequality as follows:
[tex]\[
3t + 9 \geq 0 \implies 3t \geq -9 \implies t \geq -3
\][/tex]
However, note that this inequality [tex]\(t \geq -3\)[/tex] is always true if [tex]\(t \geq 3\)[/tex]. Therefore, it doesn't add any new constraints other than what we already have from the first inequality.
3. Third Inequality:
[tex]\[
3t \geq 0
\][/tex]
This indicates that [tex]\(t\)[/tex] has to be non-negative:
[tex]\[
3t \geq 0 \implies t \geq 0
\][/tex]
This is a broader condition but also ensures that the earlier condition [tex]\(t \geq 3\)[/tex] is met.
4. Fourth Inequality:
[tex]\[
t \geq 0
\][/tex]
This is the simplest form, suggesting that [tex]\(t\)[/tex] must be non-negative to avoid any negative values for the radicand.
Conclusion:
Given all the inequalities, the minimum value for [tex]\(t\)[/tex] that satisfies all conditions and ensures the radicand is non-negative is:
[tex]\[
t \geq 0
\][/tex]
So, the answer is [tex]\(t = 0\)[/tex].
This means the smallest value [tex]\(t\)[/tex] can take without making any radicands negative is 0.
