Set up an inequality showing that the radicand cannot be negative.

A. [tex]3t - 9 \geq 0[/tex]
B. [tex]3t + 9 \geq 0[/tex]
C. [tex]3t \geq 0[/tex]
D. [tex]t \geq 0[/tex]



Answer :

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].