https://file.aiquickdraw.com/mathbot/file/1730464565197cc0og3c2.jpg

The problem here is to solve the equation \( |x + 1| = |-x - 9| \) by graphing.

### Step-by-Step Solution

To solve this equation by graphing, we’ll interpret each side of the equation as a separate function:

1. **Define the functions:**
   - Let \( f(x) = |x + 1| \)
   - Let \( g(x) = |-x - 9| \)

2. **Graph each function separately:**
   - For \( f(x) = |x + 1| \), this represents an absolute value function shifted 1 unit to the left of the origin.
   - For \( g(x) = |-x - 9| \), rewrite it as \( g(x) = |-(x + 9)| = |x + 9| \), which is an absolute value function shifted 9 units to the left.

3. **Plot these two graphs** on the same coordinate plane:
   - \( f(x) = |x + 1| \) has a "V" shape with the vertex at \( (-1, 0) \).
   - \( g(x) = |x + 9| \) has a "V" shape with the vertex at \( (-9, 0) \).

4. **Find the intersection points**:
   - The solutions to \( |x + 1| = |x + 9| \) are where the graphs of \( f(x) \) and \( g(x) \) intersect.
   - Solving algebraically or by inspection, we find that these graphs intersect when \( x = -5 \).

5. **Verification**:
   - Substitute \( x = -5 \) back into the original equation to verify:
     \[
     |x + 1| = |-x - 9|
     \]
     \[
     |-5 + 1| = |5 - 9|
     \]
     \[
     | -4 | = | -4 |
     \]
     \[
     4 = 4
     \]
   - This confirms that \( x = -5 \) is indeed a solution.

### Answer
The solution to \( |x + 1| = |-x - 9| \) is \( x = -5 \).

Would you like further clarification on any steps, or need a visual graph? 

---

Here are five related questions to deepen understanding:

1. How do we solve absolute value equations algebraically?
2. Why does the equation \( |x + 1| = |x + 9| \) simplify to only one solution?
3. What is the general process for solving an absolute value equation involving two absolute terms?
4. How can we check solutions to absolute value equations in other contexts?
5. How does shifting affect the graph of an absolute value function?

**Tip:** When solving equations involving absolute values, consider both positive and negative cases to capture all potential solutions.

Solve the equation by graphing: |x + 1| = |-x - 9|

Learn how to solve the absolute value equation |x + 1| = |-x - 9| by graphing. This problem involves graphing absolute value functions and finding intersection points, suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation