The range of x giving the solution ste of `-1lt5x+4le19` is

Options

A. `-1lexlt3`
B. `-1ltxle3`
C. `-1lexle3`
D. `-1ltxlt3`

Answer

To solve the inequality \(-1 < 5x + 4 \leq 19\), we will break it down into two parts and solve for \(x\).

### Step 1: Break down the compound inequality
We have:
\[ -1 < 5x + 4 \quad \text{and} \quad 5x + 4 \leq 19 \]

### Step 2: Solve the first part of the inequality
Starting with the first part:
\[ -1 < 5x + 4 \]
Subtract 4 from both sides:
\[ -1 - 4 < 5x \]
\[ -5 < 5x \]
Now, divide both sides by 5:
\[ -\frac{5}{5} < x \]
\[ -1 < x \quad \text{or} \quad x > -1 \]

### Step 3: Solve the second part of the inequality
Now, we solve the second part:
\[ 5x + 4 \leq 19 \]
Subtract 4 from both sides:
\[ 5x \leq 19 - 4 \]
\[ 5x \leq 15 \]
Now, divide both sides by 5:
\[ x \leq \frac{15}{5} \]
\[ x \leq 3 \]

### Step 4: Combine the results
Now we have two inequalities:
\[ x > -1 \quad \text{and} \quad x \leq 3 \]
This can be combined into a single compound inequality:
\[ -1 < x \leq 3 \]

### Final Result
The range of \(x\) that satisfies the inequality \(-1 < 5x + 4 \leq 19\) is:
\[ (-1, 3] \]