1.5x - 9 = 3x + 7. Solving Linear Equations 2. 30-4 2 4 Solving Linear Equations with Fractions 3. x²-7x+ 120 by factoring. Solving a Quadratic Equation by Factoring 4. 2x2 +3x-5 0 Solving a Quadratic Equation using the Quadratic Formula 5. 3г 2 Solving a Rational Equation 6. 2+2 +2 √x+7= x - 1. Solving a Radical Equation

Let's go through each of these problems one by one.

### 1. **Solve the linear equation:**  
\[ 1.5x - 9 = 3x + 7 \]

**Steps:**
1. Subtract \(1.5x\) from both sides:
   \[ -9 = 1.5x + 7 \]
2. Subtract 7 from both sides:
   \[ -16 = 1.5x \]
3. Divide both sides by 1.5:
   \[ x = \frac{-16}{1.5} \]
   \[ x = -\frac{32}{3} \text{ or } -10.\overline{6} \]

### 2. **Simplify the expression and solve:**  
\[ 30 - 4 \cdot 2 \cdot 4 \]

**Steps:**
1. Perform multiplication first:
   \[ 30 - 32 \]
2. Subtract:
   \[ -2 \]

### 3. **Solve the quadratic equation by factoring:**  
\[ x^2 - 7x + 12 = 0 \]

**Steps:**
1. Factor the quadratic:
   \[ (x - 3)(x - 4) = 0 \]
2. Set each factor equal to zero:
   \[ x - 3 = 0 \quad \text{or} \quad x - 4 = 0 \]
3. Solve for \(x\):
   \[ x = 3 \quad \text{or} \quad x = 4 \]

### 4. **Solve the quadratic equation using the quadratic formula:**  
\[ 2x^2 + 3x - 5 = 0 \]

**Steps:**
1. Identify the coefficients \(a = 2\), \(b = 3\), and \(c = -5\).
2. Use the quadratic formula:
   \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
   \[ x = \frac{-3 \pm \sqrt{3^2 - 4(2)(-5)}}{4} \]
   \[ x = \frac{-3 \pm \sqrt{9 + 40}}{4} \]
   \[ x = \frac{-3 \pm \sqrt{49}}{4} \]
   \[ x = \frac{-3 \pm 7}{4} \]
3. Solve for the two possible values of \(x\):
   \[ x = 1 \quad \text{or} \quad x = -\frac{5}{2} \]

### 5. **Solve the rational equation:**  
\[ \frac{3}{g} = 2 \]

**Steps:**
1. Multiply both sides by \(g\):
   \[ 3 = 2g \]
2. Divide by 2:
   \[ g = \frac{3}{2} \]

### 6. **Solve the radical equation:**  
\[ \sqrt{x+7} = x - 1 \]

**Steps:**
1. Square both sides to eliminate the square root:
   \[ x + 7 = (x - 1)^2 \]
2. Expand the right side:
   \[ x + 7 = x^2 - 2x + 1 \]
3. Rearrange to form a quadratic equation:
   \[ x^2 - 3x - 6 = 0 \]
4. Solve the quadratic equation using the quadratic formula:
   \[ x = \frac{3 \pm \sqrt{9 + 24}}{2} \]
   \[ x = \frac{3 \pm \sqrt{33}}{2} \]

The possible solutions are:
\[ x = \frac{3 + \sqrt{33}}{2} \quad \text{or} \quad x = \frac{3 - \sqrt{33}}{2} \]

Check the solutions to ensure they satisfy the original equation. Only valid real solutions will be considered.

Would you like more details on any of these problems? Here are some related questions to consider:

1. How do you verify solutions to quadratic equations?
2. What is the significance of the discriminant in the quadratic formula?
3. How do you solve equations involving absolute values?
4. What are the steps to isolate variables in multi-step equations?
5. Can you explain the process of solving inequalities?

**Tip:** Always check your solutions in the original equation, especially when dealing with radicals or fractions, to avoid extraneous solutions.

Explore examples of solving linear and quadratic equations step-by-step. Learn how to solve equations involving fractions, factoring, and radicals, with detailed explanations. Suitable for high school students and anyone preparing for math exams.

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