Example of an absolute value equation |x + 3| = |x - 11| from a math educational resource.

We are given the absolute value equation:

$$
|x + 3| = |x - 11|
$$

Our goal is to solve for $ x $. Absolute value equations can be tricky because the expression inside the absolute value can be positive or negative, so we need to consider different cases.

---

🔍 Step 1: Understand What This Equation Means

The equation says that the distance from $ x $ to $ -3 $ is equal to the distance from $ x $ to $ 11 $ on the number line.

So, geometrically, we're looking for a point $ x $ that is equidistant from $ -3 $ and $ 11 $.

That point is the midpoint between $ -3 $ and $ 11 $.

Let’s compute that:

$$
\text{Midpoint} = \frac{-3 + 11}{2} = \frac{8}{2} = 4
$$

So, $ x = 4 $ is a solution.

But let's solve it algebraically to confirm and make sure we don’t miss anything.

---

✔ Algebraic Solution

We solve:
$$
|x + 3| = |x - 11|
$$

To solve this, we use the property:
> If $ |A| = |B| $, then either $ A = B $ or $ A = -B $

So we set up two cases:

---

#### 🔹 Case 1: $ x + 3 = x - 11 $

Subtract $ x $ from both sides:
$$
3 = -11
$$

✘ This is false, so no solution in this case.

---

#### 🔹 Case 2: $ x + 3 = -(x - 11) $

Simplify the right-hand side:
$$
x + 3 = -x + 11
$$

Add $ x $ to both sides:
$$
2x + 3 = 11
$$

Subtract 3:
$$
2x = 8
$$

Divide by 2:
$$
x = 4
$$

✔ This is a valid solution.

---

🧪 Check the Solution

Plug $ x = 4 $ into the original equation:

Left side: $ |4 + 3| = |7| = 7 $
Right side: $ |4 - 11| = |-7| = 7 $

✔ Both sides are equal. So $ x = 4 $ is correct.

---

📌 Final Answer:

$$
\boxed{x = 4}
$$

This is the only solution.

---

💡 Bonus Insight: Why Only One Solution?

Because the absolute value function represents distance. The only point equidistant from $-3$ and $11$ is their midpoint — $4$. There are no other points on the number line equally distant from both.

---

Let me know if you'd like to see more example problems! 😊