For the following exercises, solve the compound inequality. Express your answer using inequality signs, and then write your answer using interval notation. [x+7<x+2]

Let's solve the compound inequality step by step.

### Given inequality:
\[
x + 7 < x + 2
\]

### Step 1: Subtract \(x\) from both sides
\[
(x + 7) - x < (x + 2) - x
\]
This simplifies to:
\[
7 < 2
\]

### Step 2: Analyze the result
The inequality \(7 < 2\) is **false**. Since there is no value of \(x\) that makes this inequality true, the solution is **no solution**.

### Answer:
There is no solution to the inequality. In interval notation, the solution is represented as the **empty set**, which can be written as:
\[
\emptyset \quad \text{or} \quad ( )
\]

Would you like more details or have any questions?

Here are 5 related questions to further your understanding:
1. How do you solve an inequality involving both "<" and "≤" signs?
2. What are the steps for solving compound inequalities with multiple parts?
3. How do you express a solution when the inequality holds for all values of \(x\)?
4. How do you handle compound inequalities involving "and" versus "or"?
5. What is the graphical interpretation of a solution with no valid \(x\)?

**Tip:** Always check if a simplified inequality results in a contradiction (like \(7 < 2\)) or a true statement (like \(7 > 2\)) to determine if there are solutions or not.

For the following exercises, solve the compound inequality. Express your answer using inequality signs, and then write your answer using interval notation. [x + 7 < x + 2]

Learn how to solve the compound inequality x + 7 < x + 2 step by step. This problem explores key algebraic concepts like inequality solving and leads to the conclusion that the solution is an empty set due to a false statement. Suitable for students in Grades 6-8.

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