Question

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

Answer 1

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:

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