Questions: Solve for (t). [ 3 t-9<6 text and -3 t<12 ]

Solve for (t). [ 3 t-9<6 text and -3 t<12 ]
Transcript text: Solve for $t$. \[ 3 t-9<6 \text { and }-3 t<12 \]
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Solution

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Solution Steps

To solve the given inequalities, we need to solve each inequality separately and then find the intersection of the solutions. The first inequality is \(3t - 9 < 6\) and the second is \(-3t < 12\). After solving these inequalities, we will determine the range of \(t\) that satisfies both conditions.

Step 1: Solve the First Inequality

The first inequality given is:

\[ 3t - 9 < 6 \]

To solve for \( t \), we first add 9 to both sides:

\[ 3t - 9 + 9 < 6 + 9 \]

\[ 3t < 15 \]

Next, divide both sides by 3:

\[ t < 5 \]

Step 2: Solve the Second Inequality

The second inequality given is:

\[ -3t < 12 \]

To solve for \( t \), divide both sides by -3. Remember that dividing by a negative number reverses the inequality sign:

\[ t > -4 \]

Step 3: Combine the Solutions

We have two inequalities:

  1. \( t < 5 \)
  2. \( t > -4 \)

Combining these, we get:

\[ -4 < t < 5 \]

Final Answer

The solution to the system of inequalities is:

\[ \boxed{-4 < t < 5} \]

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