Solve for t.

–t+14≤–2+t or t+12+2t≤12
Write your answer as a compound inequality with integers.

1 answer

To solve the compound inequality \(-t + 14 \leq -2 + t\) or \(t + 12 + 2t \leq 12\), we will solve each inequality separately.

1. Solve \(-t + 14 \leq -2 + t\):

First, rearrange the inequality:

\[ -t - t \leq -2 - 14 \]

Combine like terms:

\[ -2t \leq -16 \]

Now, divide by -2 and remember to flip the inequality sign:

\[ t \geq 8 \]

2. Solve \(t + 12 + 2t \leq 12\):

Combine like terms:

\[ 3t + 12 \leq 12 \]

Subtract 12 from both sides:

\[ 3t \leq 0 \]

Now divide by 3:

\[ t \leq 0 \]

3. Combine the results:

The results from the two parts give us:

  1. \(t \geq 8\)
  2. \(t \leq 0\)

Since we have an "or" condition, the solution is:

\[ t \leq 0 \quad \text{or} \quad t \geq 8 \]

In interval notation, this can be expressed as:

\[ (-\infty, 0] \cup [8, \infty) \]

In terms of integer solutions, we can state that:

\[ t \in { \ldots, -3, -2, -1, 0} \cup {8, 9, 10, \ldots} \]

Thus, the final solution in terms of integers is:

\[ t \leq 0 \quad \text{or} \quad t \geq 8 \]