\begin{tabular}{|l|l|l|} \hline 4.2 & Solve the lollowing equations \ \hline & 4.2 .1 & $$ 3 x+7=-1 $$ \ \hline \end{tabular} \begin{tabular}{l|l|l|} \hline 422 & $$ 2^{x}=65 $$ \ \hline \end{tabular} \begin{tabular}{|c|c} 4.2 .3 & $$ 5^{1 x+1}=25^{3-x} $$ \ \hline \end{tabular} $$ 4.2 .4 \left\lvert\, \frac{x-1}{4}-\frac{2 x-9}{7}=0 ight. $$

Question

\begin{tabular}{|l|l|l|} \hline 4.2 & Solve the lollowing equations \ \hline & 4.2 .1 & $$ 3 x+7=-1 $$ \ \hline \end{tabular} \begin{tabular}{l|l|l|} \hline 422 & $$ 2^{x}=65 $$ \ \hline \end{tabular} \begin{tabular}{|c|c} 4.2 .3 & $$ 5^{1 x+1}=25^{3-x} $$ \ \hline \end{tabular} $$ 4.2 .4 \left\lvert\, \frac{x-1}{4}-\frac{2 x-9}{7}=0ight. $$

UpStudy AI · Accepted Answer

Let's solve each of the given equations step by step.

---

### **4.2.1**  
**Equation:**  
$$ 3x + 7 = -1 $$

**Solution:**  
1. **Subtract 7 from both sides** to isolate the term with $$ x $$:
   $$
   3x + 7 - 7 = -1 - 7 \
   3x = -8
   $$
2. **Divide both sides by 3** to solve for $$ x $$:
   $$
   x = \frac{-8}{3}
   $$

**Answer:**  
$$ x = -\frac{8}{3} $$

---

### **4.2.2**  
**Equation:**  
$$ 2^{x} = 65 $$

**Solution:**  
To solve for $$ x $$, we'll use logarithms.

1. **Take the natural logarithm** of both sides:
   $$
   \ln(2^{x}) = \ln(65)
   $$
2. **Apply the power rule of logarithms** ($$ \ln(a^b) = b \ln(a) $$):
   $$
   x \ln(2) = \ln(65)
   $$
3. **Solve for $$ x $$**:
   $$
   x = \frac{\ln(65)}{\ln(2)} \approx \frac{4.1744}{0.6931} \approx 6.02
   $$

**Answer:**  
$$ x \approx 6.02 $$

---

### **4.2.3**  
**Equation:**  
$$ 5^{x + 1} = 25^{3 - x} $$

**Solution:**  
First, express both sides with the same base.

1. **Recognize that 25 is $$ 5^2 $$**:
   $$
   5^{x + 1} = (5^2)^{3 - x}
   $$
2. **Simplify the right side** using the power of a power rule ($$ (a^b)^c = a^{bc} $$):
   $$
   5^{x + 1} = 5^{2(3 - x)} = 5^{6 - 2x}
   $$
3. **Since the bases are the same, set the exponents equal**:
   $$
   x + 1 = 6 - 2x
   $$
4. **Solve for $$ x $$**:
   $$
   x + 2x = 6 - 1 \
   3x = 5 \
   x = \frac{5}{3} \approx 1.\overline{6}
   $$

**Answer:**  
$$ x = \frac{5}{3} $$

---

### **4.2.4**  
**Equation:**  
$$ \frac{x - 1}{4} - \frac{2x - 9}{7} = 0 $$

**Solution:**  
1. **Find a common denominator** to combine the fractions. The least common multiple of 4 and 7 is 28.
   $$
   \frac{7(x - 1)}{28} - \frac{4(2x - 9)}{28} = 0
   $$
2. **Combine the fractions**:
   $$
   \frac{7(x - 1) - 4(2x - 9)}{28} = 0
   $$
3. **Simplify the numerator**:
   $$
   7x - 7 - 8x + 36 = -x + 29
   $$
4. **Set the numerator equal to zero** (since the denominator is not zero):
   $$
   -x + 29 = 0 \
   x = 29
   $$

**Answer:**  
$$ x = 29 $$

---

## **Summary of Solutions:**

1. **4.2.1:** $$ x = -\dfrac{8}{3} $$
2. **4.2.2:** $$ x \approx 6.02 $$
3. **4.2.3:** $$ x = \dfrac{5}{3} $$
4. **4.2.4:** $$ x = 29 $$
Solutions to the equations are: 1. $$ x = -\frac{8}{3} $$ 2. $$ x \approx 6.02 $$ 3. $$ x = \frac{5}{3} $$ 4. $$ x = 29 $$

4.2 Solve the lollowing equations

4.2 .1

422

4.2 .3

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Solución

4.2.1

4.2.2

4.2.3

4.2.4

Summary of Solutions:

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4.2 Solve the lollowing equations 4.2 .1 422 4.2 .3