Solve the absolute value equation. $$ |2 x-7|=14 $$

Question

UpStudy AI · Accepted Answer

To solve the absolute value equation:

$$
|2x - 7| = 14
$$

We need to consider two scenarios because the absolute value of a number is its distance from zero, which means it can be positive or negative.

### Scenario 1: $$2x - 7 = 14$$

1. **Add 7 to both sides:**
   $$
   2x = 21
   $$
2. **Divide both sides by 2:**
   $$
   x = \frac{21}{2} \quad \text{or} \quad x = 10.5
   $$

### Scenario 2: $$2x - 7 = -14$$

1. **Add 7 to both sides:**
   $$
   2x = -7
   $$
2. **Divide both sides by 2:**
   $$
   x = \frac{-7}{2} \quad \text{or} \quad x = -3.5
   $$

### Final Solutions

The equation has two solutions:

$$
x = \frac{21}{2} \quad \text{and} \quad x = -\frac{7}{2}
$$

In decimal form, these are $$x = 10.5$$ and $$x = -3.5$$.

**Answer:**  
All real solutions are x = 21⁄2 and x = –7⁄2.
The solutions are $$ x = 10.5 $$ and $$ x = -3.5 $$.

Solve the absolute value equation. \[ |2 x-7|=14 \]