2+4x<2x-5<=3x, x is the element of integer 

Let's solve the compound inequality step by step:

The given inequality is:
\[
2 + 4x < 2x - 5 \leq 3x
\]
We will break this into two separate inequalities:

1. \( 2 + 4x < 2x - 5 \)
2. \( 2x - 5 \leq 3x \)

### Step 1: Solve the first inequality \( 2 + 4x < 2x - 5 \)
\[
2 + 4x < 2x - 5
\]
Subtract \( 2x \) from both sides:
\[
2 + 2x < -5
\]
Subtract 2 from both sides:
\[
2x < -7
\]
Divide by 2:
\[
x < -\frac{7}{2}
\]
Since \( x \) is an integer, this simplifies to:
\[
x < -4
\]

### Step 2: Solve the second inequality \( 2x - 5 \leq 3x \)
\[
2x - 5 \leq 3x
\]
Subtract \( 2x \) from both sides:
\[
-5 \leq x
\]
This simplifies to:
\[
x \geq -5
\]

### Step 3: Combine the two inequalities
We now have the following inequalities:
\[
x < -4 \quad \text{and} \quad x \geq -5
\]
The solution is \( -5 \leq x < -4 \), meaning \( x = -5 \) is the only integer solution.

### Final Answer:
The integer solution is:
\[
x = -5
\]

Would you like more details, or do you have any questions?

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Here are five related questions you can explore:

1. What is the solution for \( 3x + 7 < 2x - 4 \leq 5x + 6 \)?
2. How do you solve inequalities involving absolute values?
3. Can you graph the solution of \( 2x + 1 \leq 5x - 3 \) on a number line?
4. How does solving a compound inequality differ from solving a simple inequality?
5. What happens if the solution of an inequality involves fractions?

**Tip:** When solving inequalities, always remember to reverse the inequality sign if you multiply or divide both sides by a negative number!

2 + 4x < 2x - 5 <= 3x, x is the element of integer

This page provides a step-by-step solution to the compound inequality 2 + 4x < 2x - 5 <= 3x. Learn how to solve inequalities by breaking them into simpler parts and applying algebraic principles. Suitable for students in Grades 8-10.

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