Solve the following system of inequations:
`5x+1gt-24,5x-1lt24`.

Answer

To solve the system of inequations \( 5x + 1 > -24 \) and \( 5x - 1 < 24 \), we will tackle each inequation step by step.

### Step 1: Solve the first inequation \( 5x + 1 > -24 \)

1. **Subtract 1 from both sides:**
\[ 5x + 1 - 1 > -24 - 1 \]
This simplifies to:
\[ 5x > -25 \]

2. **Divide both sides by 5:**
\[ x > -5 \]

### Step 2: Solve the second inequation \( 5x - 1 < 24 \)

1. **Add 1 to both sides:**
\[ 5x - 1 + 1 < 24 + 1 \]
This simplifies to:
\[ 5x < 25 \]

2. **Divide both sides by 5:**
\[ x < 5 \]

### Step 3: Combine the results

From the first inequation, we have:
\[ x > -5 \]
From the second inequation, we have:
\[ x < 5 \]

### Step 4: Write the solution in interval notation

The solution to the system of inequations is:
\[ -5 < x < 5 \]
This can be expressed in interval notation as:
\[ (-5, 5) \]

### Summary of the Solution
The solution to the given system of inequations is:
\[ (-5, 5) \]