$$ 2 \frac { 1 } { 4 } = \frac { x } { 2 } + t \leq 1 \frac { 1 } { 6 } $$

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To solve the inequality $$ 2 \frac{1}{4} = \frac{x}{2} + t \leq 1 \frac{1}{6} $$, we will break it down into steps.

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we need to convert the mixed numbers into improper fractions.

1. $$ 2 \frac{1}{4} = 2 \times 4 + 1 = 8 + 1 = 9 $$ so $$ 2 \frac{1}{4} = \frac{9}{4} $$.
2. $$ 1 \frac{1}{6} = 1 \times 6 + 1 = 6 + 1 = 7 $$ so $$ 1 \frac{1}{6} = \frac{7}{6} $$.

Now, we can rewrite the equation and inequality:
$$
\frac{9}{4} = \frac{x}{2} + t \leq \frac{7}{6}
$$

### Step 2: Solve the Equation
We will first solve the equation $$ \frac{9}{4} = \frac{x}{2} + t $$.

Rearranging gives:
$$
t = \frac{9}{4} - \frac{x}{2}
$$

### Step 3: Solve the Inequality
Next, we will solve the inequality $$ \frac{x}{2} + t \leq \frac{7}{6} $$.

Substituting $$ t $$ from the equation into the inequality:
$$
\frac{x}{2} + \left( \frac{9}{4} - \frac{x}{2} \right) \leq \frac{7}{6}
$$

This simplifies to:
$$
\frac{9}{4} \leq \frac{7}{6}
$$

### Step 4: Check the Inequality
Now we need to check if $$ \frac{9}{4} \leq \frac{7}{6} $$.

To compare these fractions, we can find a common denominator. The least common multiple of 4 and 6 is 12.

Converting both fractions:
- $$ \frac{9}{4} = \frac{9 \times 3}{4 \times 3} = \frac{27}{12} $$
- $$ \frac{7}{6} = \frac{7 \times 2}{6 \times 2} = \frac{14}{12} $$

Now we compare:
$$
\frac{27}{12} \leq \frac{14}{12}
$$

This is false, so the inequality does not hold.

### Conclusion
The equation $$ \frac{9}{4} = \frac{x}{2} + t $$ can be solved for $$ t $$ in terms of $$ x $$, but the inequality $$ \frac{x}{2} + t \leq \frac{7}{6} $$ does not hold true under the given conditions.

Thus, there are no values of $$ x $$ and $$ t $$ that satisfy both the equation and the inequality simultaneously.
There are no values of $$ x $$ and $$ t $$ that satisfy both the equation and the inequality simultaneously.

\( 2 \frac { 1 } { 4 } = \frac { x } { 2 } + t \leq 1 \frac { 1 } { 6 } \)

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