Problem 23 - Algebra A Combined Function

Question

Solve each equation. See Examples 3 through \(5 .\) $$ \frac{2(x+1)}{4}=3 x-2 $$

Step-by-Step Solution

Answer

The solution is \( x = 1 \).

Step 1: Clear the Fraction

To eliminate the fraction, multiply every term in the equation by 4, the denominator. This gives: \[ 4 \times \frac{2(x+1)}{4} = 4 \times (3x - 2) \]Simplifying both sides, we get:\[ 2(x+1) = 12x - 8 \]

Step 2: Distribute the Left Side

Distribute the 2 in the expression \(2(x+1)\) on the left side:\[ 2 \times x + 2 \times 1 = 12x - 8 \]This simplifies to:\[ 2x + 2 = 12x - 8 \]

Step 3: Move Terms with x to One Side

Subtract \(2x\) from both sides to gather all the \(x\) terms on one side of the equation:\[ 2x + 2 - 2x = 12x - 8 - 2x \]This simplifies to:\[ 2 = 10x - 8 \]

Step 4: Isolate the Constant

Add 8 to both sides in order to move the constant term to the other side:\[ 2 + 8 = 10x - 8 + 8 \]Which simplifies to:\[ 10 = 10x \]

Step 5: Solve for x

Divide both sides by 10 to solve for \(x\):\[ \frac{10}{10} = \frac{10x}{10} \]Simplifying gives:\[ x = 1 \]

Key Concepts

Clear the Fraction

Fractions in equations can often appear daunting. However, by learning the technique to clear them, we can make the equation much simpler to solve. To clear the fraction in the equation \( \frac{2(x+1)}{4} = 3x - 2 \), we need to eliminate the denominator, which is 4 in this case. To achieve this, we multiply every term within the equation by 4. By doing so, the fractional part of the equation will be resolved, turning it into a standard equation that's easier to handle.

Here's how you do it:

• Multiply each term by the denominator, so: \( 4 \times \frac{2(x+1)}{4} \) eliminates the fraction to give \( 2(x+1) \).
• The right side also gets multiplied by 4, producing \( 4 \times (3x - 2) = 12x - 8 \).

This is a crucial step that simplifies complex algebra problems involving fractions.

Distribute the Expression

Once we've cleared the fraction, the next step is to simplify the equation further by distributing any expressions. In our example, the left side has the expression \( 2(x+1) \). The process of distributing involves applying the multiplication across the terms within the parentheses.

Here's what you do:

• Multiply 2 by each term inside the parenthesis: \( 2 \times x \) gives \( 2x \), and \( 2 \times 1 \) gives \( 2 \).

So, \( 2(x+1) \) simplifies to \( 2x + 2 \).
The result is a simplified, linear equation \( 2x + 2 = 12x - 8 \) where terms can be further moved and manipulated more easily. Distributing expressions is essential in breaking down and simplifying complex equations, paving the way for simpler manipulations in further steps.

Isolate the Variable

After we distribute the expressions and simplify both sides of the equation, the ultimate goal is to solve for the variable, usually \( x \). This entails isolating the variable on one side of the equation. The equation after distribution is \( 2x + 2 = 12x - 8 \). Moving all terms with \( x \) to one side requires a bit of basic algebraic manipulation.

To isolate the variable:

• Subtract \( 2x \) from both sides to keep all \( x \)-related terms on the same side: \( 12x - 2x = 10x \).
• The equation becomes \( 2 = 10x - 8 \).
• Next, add 8 to both sides to get \( 10 = 10x \).
• Finally, divide both sides by 10 to isolate \( x \), giving \( x = 1 \).

Isolating the variable ensures that \( x \) is expressed clearly and that the solution of the equation is evident. This is a crucial skill in solving algebraic equations efficiently.