Problem 8 - College Algebra

Question

For the following exercises, solve the equation for \(x\). $$ 3(x+2)-12=5(x+1) $$

Step-by-Step Solution

Answer

The solution is \(x = -\frac{11}{2}\).

Step 1: Distribute terms inside the parentheses

Distribute the 3 to both terms inside the parentheses on the left side of the equation: \[ 3(x+2) \Rightarrow 3x + 6 \]and distribute the 5 on the right side: \[ 5(x+1) \Rightarrow 5x + 5 \]

Step 2: Rewrite the equation

Substitute the distributed terms back into the original equation: \[ 3x + 6 - 12 = 5x + 5 \]

Step 3: Simplify the equation

Simplify the left side by combining like terms: \[ 3x + 6 - 12 \Rightarrow 3x - 6 \]The equation now looks like: \[ 3x - 6 = 5x + 5 \]

Step 4: Move the variable terms to one side

Subtract \(5x\) from both sides to get all \(x\) terms on one side of the equation: \[ 3x - 5x - 6 = 5x - 5x + 5 \]This simplifies to: \[-2x - 6 = 5 \]

Step 5: Move constant terms to the other side

Add 6 to both sides to isolate the \(x\) term: \[-2x - 6 + 6 = 5 + 6 \]Which simplifies to: \[-2x = 11 \]

Step 6: Solve for x

Divide both sides by \(-2\) to solve for \(x\): \[ x = \frac{11}{-2} \]Simplifying gives: \[ x = -\frac{11}{2} \]

Key Concepts

Distributive Property

The distributive property is a vital principle often used in algebra, and it's especially helpful when solving equations involving parentheses. It allows us to multiply a single term by each term inside a parenthesis. This is necessary in solving linear equations where terms are grouped. In our example, the equation is initially \(3(x+2)-12=5(x+1)\).
Let's apply the distributive property:

• Multiply 3 by each term inside the first parenthesis: \(3(x+2) \to 3 \cdot x + 3 \cdot 2 = 3x + 6\).
• Now, do the same on the right side with 5: \(5(x+1) \to 5 \cdot x + 5 \cdot 1 = 5x + 5\).

After distributing, we convert the equation to: \(3x + 6 - 12 = 5x + 5\). Remember, practicing the distributive property will make it second nature as you solve more equations.

Combining Like Terms

Combining like terms is the process of simplifying expressions by merging terms that have identical variable parts. This simplification makes it easier to solve the equation.
In our example: \(3x + 6 - 12\), we can combine the numbers:

• Look at the constants: \(6 - 12\). Adding these gives us \(-6\).

This combination of like terms simplifies the equation to: \(3x - 6 = 5x + 5\). By ensuring we have combined all possible like terms, we reduce the complexity of the equation, making each subsequent step simpler to manage.

Isolating Variables

To isolate the variable means to get the variable alone on one side of the equation. This is key to solving for that variable.
Let's start with the simplified equation \(3x - 6 = 5x + 5\). Here are the steps to isolate \(x\):

• Subtract \(5x\) from both sides to focus the variable terms on one side: \(3x - 5x - 6 = 5\).
• This operation gives: \(-2x - 6 = 5\).
• Add 6 to both sides to separate the \(x\) term: \(-2x = 11\).

Isolating variables through operations like addition, subtraction, multiplication, and division helps simplify to a point where the solution is straightforward.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations that represent specific values. Understanding them is crucial in every step of solving equations.
In our example, we worked with expressions like \(3(x+2)\) and later transformed it into expressions without parentheses through distribution.
Algebraic expressions help us to represent real situations in abstract forms.
They allow us to use mathematical operations to manipulate these expressions to find unknown values, like \(x\) in our problem.
Understanding the parts of algebraic expressions will make complex math procedures feel more intuitive.