Certainly! Let's break down the problem step-by-step to set up the inequality such that the radicand (the expression inside a square root) cannot be negative.
1. Understanding the Radicand:
The radicand is the expression under the square root. For a square root to be defined in the real numbers, the radicand must be non-negative, which means it must be greater than or equal to zero.
2. Setting Up the Inequalities:
Here we have four expressions that need to be non-negative. Let's analyze each one to determine the appropriate inequalities:
- Expression [tex]\(3t - 9 \geq 0\)[/tex]:
We start with:
[tex]\[ 3t - 9 \geq 0 \][/tex]
Adding 9 to both sides gives:
[tex]\[ 3t \geq 9 \][/tex]
Dividing both sides by 3 results in:
[tex]\[ t \geq 3 \][/tex]
- Expression [tex]\(3t + 9 \geq 0\)[/tex]:
Starting with:
[tex]\[ 3t + 9 \geq 0 \][/tex]
Subtracting 9 from both sides gives:
[tex]\[ 3t \geq -9 \][/tex]
Dividing both sides by 3 results in:
[tex]\[ t \geq -3 \][/tex]
- Expression [tex]\(3t \geq 0\)[/tex]:
This inequality states:
[tex]\[ 3t \geq 0 \][/tex]
Dividing both sides by 3 gives:
[tex]\[ t \geq 0 \][/tex]
- Expression [tex]\(t \geq 0\)[/tex]:
This inequality is already in its simplest form:
[tex]\[ t \geq 0 \][/tex]
3. Combining the Inequalities:
With the inequalities derived, we have:
[tex]\[
t \geq 3
\][/tex]
[tex]\[
t \geq -3
\][/tex]
[tex]\[
t \geq 0
\][/tex]
[tex]\[
t \geq 0
\][/tex]
Among these, the most restrictive condition is [tex]\( t \geq 3 \)[/tex] because if [tex]\( t \geq 3 \)[/tex] is satisfied, the other conditions ([tex]\( t \geq -3 \)[/tex], [tex]\( t \geq 0 \)[/tex]) will automatically be satisfied.
So, to ensure that the radicand is non-negative for all these expressions, the final condition that satisfies all individual inequalities is:
[tex]\[ t \geq 3 \][/tex]
This ensures that all our expressions under consideration are non-negative.
