Questions: Solve the linear inequality, and write your answer in interval notation. (2+t)/5-(t+4)/3 > -1/15

Solution Steps

To solve the linear inequality, we need to follow these steps:

1. Eliminate the fractions by finding a common denominator.
2. Combine like terms and simplify the inequality.
3. Solve for the variable \( t \).
4. Write the solution in interval notation.

To eliminate the fractions, we find a common denominator for the terms in the inequality: \[ \frac{2+t}{5} - \frac{t+4}{3} > -\frac{1}{15} \] The common denominator for 5, 3, and 15 is 15. Multiplying each term by 15, we get: \[ 15 \left( \frac{2+t}{5} \right) - 15 \left( \frac{t+4}{3} \right) > 15 \left( -\frac{1}{15} \right) \] Simplifying, we obtain: \[ 3(2+t) - 5(t+4) > -1 \]

Next, we distribute and combine like terms: \[ 3(2+t) - 5(t+4) > -1 \] \[ 6 + 3t - 5t - 20 > -1 \] \[ 6 - 2t - 20 > -1 \] \[ -2t - 14 > -1 \]

Isolate \( t \) by adding 14 to both sides and then dividing by -2: \[ -2t > 13 \] \[ t < -6.5 \]

Final Answer