https://file.aiquickdraw.com/mathbot/file/1728919110453t4l0gnkc.jpg

Let's solve the compound inequality:

\[
3x + 2 < 5x + 4 \leq 10x - 1
\]

This is a double inequality, which we can break into two parts:

### 1. Solve \( 3x + 2 < 5x + 4 \):
Subtract \( 3x \) from both sides:

\[
2 < 2x + 4
\]

Now subtract 4 from both sides:

\[
2 - 4 < 2x
\]
\[
-2 < 2x
\]

Divide by 2:

\[
-1 < x
\]

So, \( x > -1 \).

### 2. Solve \( 5x + 4 \leq 10x - 1 \):
Subtract \( 5x \) from both sides:

\[
4 \leq 5x - 1
\]

Now add 1 to both sides:

\[
5 \leq 5x
\]

Divide by 5:

\[
1 \leq x
\]

So, \( x \geq 1 \).

### Combine the results:
From both inequalities, we have \( x > -1 \) and \( x \geq 1 \). Therefore, the solution is:

\[
x \geq 1
\]

### Interval notation:
The solution in interval form is:

\[
[1, \infty)
\]

Would you like further details or have any questions?

Here are some related questions to expand on this:

1. How do you solve a compound inequality with both strict and non-strict inequalities?
2. What is the difference between solving inequalities and equations?
3. How does solving a compound inequality differ from solving a single inequality?
4. What are the key steps in expressing solutions in interval notation?
5. How do you interpret strict inequalities (like \( < \) or \( > \)) in interval notation?

**Tip:** Always double-check your interval endpoints, especially when dealing with strict vs. non-strict inequalities, to avoid minor errors.

Solve the following linear inequality and state the answer in interval form: 3x + 2 < 5x + 4 ≤ 10x - 1

Solve the compound linear inequality 3x + 2 < 5x + 4 ≤ 10x - 1 and express the solution in interval form. This example covers essential algebraic concepts, including solving and combining inequalities. Suitable for students in Grades 9-10.

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