Solution set for inequality `-8 le5x - 3 < 7` is (-1,2).

Answer

To solve the inequality \(-8 \leq 5x - 3 < 7\), we will break it down step by step.

### Step 1: Split the compound inequality
We can rewrite the compound inequality as two separate inequalities:
1. \(-8 \leq 5x - 3\)
2. \(5x - 3 < 7\)

### Step 2: Solve the first inequality
Starting with the first inequality:
\[ -8 \leq 5x - 3 \]
Add 3 to both sides:
\[ -8 + 3 \leq 5x \]
This simplifies to:
\[ -5 \leq 5x \]
Now, divide both sides by 5:
\[ -\frac{5}{5} \leq x \]
This simplifies to:
\[ -1 \leq x \quad \text{or} \quad x \geq -1 \]

### Step 3: Solve the second inequality
Now, let's solve the second inequality:
\[ 5x - 3 < 7 \]
Add 3 to both sides:
\[ 5x < 7 + 3 \]
This simplifies to:
\[ 5x < 10 \]
Now, divide both sides by 5:
\[ x < \frac{10}{5} \]
This simplifies to:
\[ x < 2 \]

### Step 4: Combine the results
From the two inequalities we have:
1. \(x \geq -1\)
2. \(x < 2\)

Combining these results gives us:
\[ -1 \leq x < 2 \]

### Step 5: Write the solution set
The solution set in interval notation is:
\[ [-1, 2) \]

### Final Answer
The solution set for the inequality \(-8 \leq 5x - 3 < 7\) is \([-1, 2)\).

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