Find the range of vaues of x, which satisfy the inequality :
`-1/5 le (3x)/(10 ) + 1 lt (2)/(5) , x in R.`
Graph the solution set on the number line.

Answer

To solve the inequality \(-\frac{1}{5} \leq \frac{3x}{10} + 1 < \frac{2}{5}\), we will break it down into two parts and solve each part step by step.

### Step 1: Break down the inequality
We can rewrite the compound inequality as two separate inequalities:
1. \(-\frac{1}{5} \leq \frac{3x}{10} + 1\)
2. \(\frac{3x}{10} + 1 < \frac{2}{5}\)

### Step 2: Solve the first inequality
Starting with the first inequality:
\[ -\frac{1}{5} \leq \frac{3x}{10} + 1 \]
Subtract 1 from both sides:
\[ -\frac{1}{5} - 1 \leq \frac{3x}{10} \]
Convert 1 to a fraction with a denominator of 5:
\[ -\frac{1}{5} - \frac{5}{5} = -\frac{6}{5} \]
So we have:
\[ -\frac{6}{5} \leq \frac{3x}{10} \]
Now, multiply both sides by 10 to eliminate the fraction:
\[ 10 \cdot -\frac{6}{5} \leq 3x \]
This simplifies to:
\[ -12 \leq 3x \]
Now, divide both sides by 3:
\[ -4 \leq x \quad \text{or} \quad x \geq -4 \]

### Step 3: Solve the second inequality
Now, we will solve the second inequality:
\[ \frac{3x}{10} + 1 < \frac{2}{5} \]
Subtract 1 from both sides:
\[ \frac{3x}{10} < \frac{2}{5} - 1 \]
Convert 1 to a fraction with a denominator of 5:
\[ \frac{2}{5} - \frac{5}{5} = -\frac{3}{5} \]
So we have:
\[ \frac{3x}{10} < -\frac{3}{5} \]
Now, multiply both sides by 10:
\[ 3x < 10 \cdot -\frac{3}{5} \]
This simplifies to:
\[ 3x < -6 \]
Now, divide both sides by 3:
\[ x < -2 \]

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

Combining these gives us:
\[ -4 \leq x < -2 \]

### Step 5: Write the final answer
The range of values of \(x\) that satisfy the inequality is:
\[ x \in [-4, -2) \]

### Step 6: Graph the solution set on the number line
To represent this on a number line:
- Mark the point \(-4\) with a closed dot (indicating that \(-4\) is included).
- Mark the point \(-2\) with an open dot (indicating that \(-2\) is not included).
- Shade the region between \(-4\) and \(-2\).