Solve the inequality. Describe the relationship between the solution and the graphs of the individual inequalities. 5 is less than or equal to x + 7 is less than or equal to 12. Enter the solutions in the blanks. BLANK OR BLANK

The given inequality is:

\[ 5 \leq x + 7 \leq 12 \]

This is a compound inequality, meaning we have two inequalities combined into one. We need to solve for \( x \) by breaking it down into two separate inequalities.

### Step 1: Solve the Left Inequality
The first part of the inequality is:

\[ 5 \leq x + 7 \]

To solve for \( x \), subtract 7 from both sides:

\[ 5 - 7 \leq x \]
\[ -2 \leq x \]
\[ x \geq -2 \]

### Step 2: Solve the Right Inequality
The second part of the inequality is:

\[ x + 7 \leq 12 \]

To solve for \( x \), subtract 7 from both sides:

\[ x \leq 12 - 7 \]
\[ x \leq 5 \]

### Step 3: Combine the Results
The solution to the compound inequality is:

\[ -2 \leq x \leq 5 \]

This means \( x \) must be greater than or equal to \(-2\) and less than or equal to 5.

### Graphical Interpretation
- The graph of \( x \geq -2 \) is a ray starting from \( x = -2 \) and extending to the right.
- The graph of \( x \leq 5 \) is a ray starting from \( x = 5 \) and extending to the left.

The solution to the combined inequality \( -2 \leq x \leq 5 \) represents the intersection of these two rays, which is the segment from \( -2 \) to \( 5 \) on the number line.

### Final Solution
The solution can be written as:
**" \( -2 \leq x \leq 5 \)"**

This should be entered as **" \( -2 \leq x \leq 5 \)"**.

Would you like more details on this solution or have any questions?

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Here are 5 related questions to expand on this concept:
1. How would the solution change if one of the inequalities was strict, such as \( 5 < x + 7 \leq 12 \)?
2. What would the solution be if the compound inequality was \( 7 \leq x + 7 \leq 14 \)?
3. Can you describe the graph of the inequality \( x - 3 \leq 9 \) and how it relates to the solution?
4. What happens to the solution set if the inequalities are reversed, like \( 12 \leq x + 7 \leq 15 \)?
5. How do you solve and graph an inequality with three terms, like \( 1 \leq 2x + 3 \leq 7 \)?

**Tip:** When solving compound inequalities, treat each part separately and then combine the results to determine the range where both conditions are true.

Learn how to solve the compound inequality 5 ≤ x + 7 ≤ 12 step-by-step with detailed explanations. Understand the relationship between the solution and the graphical representation of individual inequalities. Suitable for students in Grades 7-9.

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