Solve the equation $$ |2 x+1|=10 $$

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UpStudy AI · Accepted Answer

To solve the equation $$ |2x + 1| = 10 $$, we'll consider the definition of absolute value. The absolute value of an expression equals a positive number, which means the expression inside the absolute value can be either positive or negative. Therefore, we can set up two separate equations:

1. **Case 1:** $$ 2x + 1 = 10 $$
2. **Case 2:** $$ 2x + 1 = -10 $$

Let's solve each case step by step.

---

### **Case 1:** $$ 2x + 1 = 10 $$

1. **Subtract 1 from both sides:**
   $$
   2x = 10 - 1
   $$
   $$
   2x = 9
   $$

2. **Divide both sides by 2:**
   $$
   x = \frac{9}{2}
   $$
   $$
   x = 4.5
   $$

---

### **Case 2:** $$ 2x + 1 = -10 $$

1. **Subtract 1 from both sides:**
   $$
   2x = -10 - 1
   $$
   $$
   2x = -11
   $$

2. **Divide both sides by 2:**
   $$
   x = \frac{-11}{2}
   $$
   $$
   x = -5.5
   $$

---

### **Conclusion**

The solutions to the equation $$ |2x + 1| = 10 $$ are:
$$
x = \frac{9}{2} \quad \text{and} \quad x = -\frac{11}{2}
$$
Or in decimal form:
$$
x = 4.5 \quad \text{and} \quad x = -5.5
$$

You can verify these solutions by plugging them back into the original equation:

- **For $$ x = 4.5 $$:**
  $$
  |2(4.5) + 1| = |9 + 1| = |10| = 10
  $$
  
- **For $$ x = -5.5 $$:**
  $$
  |2(-5.5) + 1| = |-11 + 1| = |-10| = 10
  $$

Both solutions satisfy the original equation.
The solutions are $$ x = 4.5 $$ and $$ x = -5.5 $$.

Solve the equation

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