Question

Find the range of values of x which satisfies the inequality 3x - 2 < 10 + x < 2 + 5x

Answer 1

To solve the compound inequality \( 3x - 2 < 10 + x < 2 + 5x \), we can break it down into two parts:

1. \( 3x - 2 < 10 + x \)
2. \( 10 + x < 2 + 5x \)

### Step 1: Solve the first inequality \( 3x - 2 < 10 + x \)

Start by isolating \( x \):

\[
3x - 2 < 10 + x
\]

Subtract \( x \) from both sides:

\[
3x - x - 2 < 10
\]

This simplifies to:

\[
2x - 2 < 10
\]

Next, add 2 to both sides:

\[
2x < 12
\]

Finally, divide by 2:

\[
x < 6
\]

### Step 2: Solve the second inequality \( 10 + x < 2 + 5x \)

Again, isolate \( x \):

\[
10 + x < 2 + 5x
\]

Subtract \( x \) from both sides:

\[
10 < 2 + 5x - x
\]

This simplifies to:

\[
10 < 2 + 4x
\]

Now, subtract 2 from both sides:

\[
8 < 4x
\]

Finally, divide by 4:

\[
2 < x \quad \text{or} \quad x > 2
\]

### Step 3: Combine the results

From the two parts, we have:

1. \( x < 6 \)
2. \( x > 2 \)

Thus, the combined solution is:

\[
2 < x < 6
\]

### Conclusion

The range of values for \( x \) that satisfies the original inequality is:

\[
\boxed{(2, 6)}
\]