Solve for x 3(x+2)=(5)

Solution:

Step 1: Isolate the variable term

Isolate the $x$ term by dividing the equation $3(x+2) = 5$ by $3$.

Step 1.1: Apply the division to both sides

Apply division to both sides: $\frac{3(x+2)}{3} = \frac{5}{3}$.

Step 1.2: Simplify the equation

Simplify the equation by removing common factors.

Step 1.2.1: Eliminate the common factor

Eliminate the common factor of $3$: $\frac{\cancel{3}(x+2)}{\cancel{3}} = \frac{5}{3}$.

Step 1.2.1.1: Simplify the left side

Simplify the left side: $x + 2 = \frac{5}{3}$.

Step 2: Move constant terms to the other side

Move the constant term to the other side to isolate $x$.

Step 2.1: Subtract the constant from both sides

Subtract $2$ from both sides: $x = \frac{5}{3} - 2$.

Step 2.2: Convert the constant to a fraction

Convert $-2$ to a fraction with a common denominator: $x = \frac{5}{3} - 2 \cdot \frac{3}{3}$.

Step 2.3: Combine the fractions

Combine the fractions: $x = \frac{5}{3} - \frac{6}{3}$.

Step 2.4: Add the numerators

Add the numerators over the common denominator: $x = \frac{5 - 6}{3}$.

Step 2.5: Simplify the numerator

Simplify the numerator by performing the subtraction.

Step 2.5.1: Perform the multiplication

Multiply $-2$ by $3$: $x = \frac{5 - 6}{3}$.

Step 2.5.2: Complete the subtraction

Complete the subtraction in the numerator: $x = \frac{-1}{3}$.

Step 2.6: Write the negative sign outside the fraction

Place the negative sign in front of the fraction: $x = -\frac{1}{3}$.

Step 3: Express the result in different forms

The solution can be expressed in various forms.

Exact Form: $x = -\frac{1}{3}$

Decimal Form: $x = -0.333...$

Knowledge Notes:

To solve the equation $3(x+2) = 5$, we follow a systematic approach:

1. Distributive Property: This property allows us to multiply a number by a sum by multiplying each addend separately and then summing the products. In this case, we start by dividing both sides by $3$ to simplify the equation.

2. Simplifying Equations: Simplifying involves reducing the complexity of an expression or equation. We do this by canceling out common factors and combining like terms.

3. Fraction Arithmetic: When dealing with fractions, we aim to have a common denominator to combine terms. This often involves converting whole numbers to fractions.

4. Isolating the Variable: The goal in solving an equation is to isolate the variable on one side of the equation. We do this by performing the same operation on both sides of the equation.

5. Negative Numbers and Fractions: When dealing with negative numbers, especially in fractions, it is important to correctly place the negative sign. It can be placed in front of the numerator, the denominator, or in front of the entire fraction, as they are equivalent.

6. Alternative Forms of the Solution: Solutions to equations can be presented in various forms, such as exact form (fractions) or decimal form. It is important to understand how to convert between these forms for accurate representation of the solution.