Solve the equation \( 2 x+4 \frac{1}{5}=9 \). Explain the steps and properties you used.

Certainly! Let's solve the equation step by step and explain each part of the process. **Given Equation:** \[ 2x + 4 \frac{1}{5} = 9 \] **Step 1: Convert the Mixed Number to an Improper Fraction** First, it's easier to work with improper fractions rather than mixed numbers. The term \( 4 \frac{1}{5} \) is a mixed number, which we can convert to an improper fraction. \[ 4 \frac{1}{5} = \left(4 \times 5\right) + 1 = 20 + 1 = 21 \Rightarrow \frac{21}{5} \] So, the equation becomes: \[ 2x + \frac{21}{5} = 9 \] **Step 2: Isolate the Variable Term** Our goal is to solve for \( x \), so we'll first isolate the term containing \( x \). To do this, subtract \( \frac{21}{5} \) from both sides of the equation. \[ 2x + \frac{21}{5} - \frac{21}{5} = 9 - \frac{21}{5} \] \[ 2x = 9 - \frac{21}{5} \] **Step 3: Simplify the Right Side** To subtract \( \frac{21}{5} \) from 9, it's helpful to express 9 as a fraction with a denominator of 5. \[ 9 = \frac{45}{5} \] \[ 2x = \frac{45}{5} - \frac{21}{5} \] \[ 2x = \frac{45 - 21}{5} \] \[ 2x = \frac{24}{5} \] **Step 4: Solve for \( x \)** Now, divide both sides of the equation by 2 to solve for \( x \). \[ 2x = \frac{24}{5} \] \[ x = \frac{24}{5} \div 2 \] \[ x = \frac{24}{5} \times \frac{1}{2} \] \[ x = \frac{24}{10} \] \[ x = \frac{12}{5} \] \[ x = 2 \frac{2}{5} \] **Solution:** \[ x = \frac{12}{5} \quad \text{or} \quad x = 2 \frac{2}{5} \] **Summary of Properties and Steps Used:** 1. **Conversion of Mixed Numbers:** Changed the mixed number \( 4 \frac{1}{5} \) to an improper fraction \( \frac{21}{5} \) for easier manipulation. 2. **Isolating the Variable:** Subtracted \( \frac{21}{5} \) from both sides to isolate the term containing \( x \). 3. **Common Denominator:** Converted whole numbers to fractions with a common denominator to perform the subtraction. 4. **Solving for \( x \):** Divided both sides by the coefficient of \( x \) (which is 2) to find the value of \( x \). By following these steps systematically, we arrive at the solution \( x = \frac{12}{5} \) or \( x = 2 \frac{2}{5} \).