Problem 1 - Algebra for College Students

Question

For Problems \(1-44\), solve each equation. $$ \frac{x+1}{4}+\frac{x-2}{6}=\frac{3}{4} $$

Step-by-Step Solution

Answer

The solution is \( x = 2 \).

Step 1: Find a Common Denominator

Before solving the equation, we need a common denominator for the fractions involved. The denominators 4 and 6 have a least common multiple (LCM) of 12. So, we will multiply each term in the equation by 12 to eliminate the fractions.

Step 2: Eliminate the Fractions

Multiply each term by 12 to eliminate the denominators:\[ 12 \left( \frac{x+1}{4} \right) + 12 \left( \frac{x-2}{6} \right) = 12 \times \frac{3}{4} \]This simplifies to:\[ 3(x+1) + 2(x-2) = 9 \]

Step 3: Distribute and Simplify

Distribute the numbers outside the parentheses:\[ 3x + 3 + 2x - 4 = 9 \]Combine like terms:\[ 5x - 1 = 9 \]

Step 4: Solve for x

Add 1 to both sides to isolate the term with x:\[ 5x = 10 \]Now, divide both sides by 5 to solve for x:\[ x = 2 \]

Step 5: Verify the Solution

Substitute \( x = 2 \) back into the original equation to ensure it holds true:Substitute into left side:\[ \frac{2+1}{4} + \frac{2-2}{6} = \frac{3}{4} \]Simplify:\[ \frac{3}{4} + 0 = \frac{3}{4} \]The left side matches the right side, confirming that \( x = 2 \) is correct.

Key Concepts

Common Denominator

When solving equations with fractions, finding a common denominator helps to eliminate fractions and simplify calculations. The common denominator is essentially the least common multiple (LCM) of all the denominators in the equation.

For example, in the equation \( \frac{x+1}{4} + \frac{x-2}{6} = \frac{3}{4} \), the denominators are 4 and 6. The LCM of these numbers is 12. Using 12 as a common denominator allows us to rewrite each term with this denominator, thus making the fractions easier to handle.

This step turns the task into working with integers by multiplying through by the common denominator, often making the equation much simpler to solve. It's crucial because it ensures that all fraction terms are comparable, avoiding mistakes in calculations.

Fractions

Fractions represent parts of a whole and are composed of a numerator and a denominator. In equations, fractions can complicate calculations, but they can be managed effectively by turning them into equivalent forms.

In our given exercise, fractions such as \( \frac{x+1}{4} \) and \( \frac{x-2}{6} \) are present. By identifying a common denominator, these fractions can be rewritten, making them easier to add or subtract.

• Multiply each fraction by the number which turns the denominators into the common denominator.
• Ensure you perform the same operation on both sides to maintain balance in the equation.
• Convert all fractions, performing all necessary operations to simplify them.

Understanding fractions is key to solving equations effectively, allowing smooth transitions into integer operations.

Distributive Property

The distributive property is a key principle in algebra that allows you to multiply a sum or difference by a number. This concept is expressed as \( a(b + c) = ab + ac \). Utilizing the distributive property can simplify complex expressions into more straightforward components.

For example, when applying this property to the equation \( 3(x+1) + 2(x-2) = 9 \), distribute the numbers 3 and 2 to each term inside the parentheses:

• \( 3 \times (x+1) \) becomes \( 3x + 3 \).
• \( 2 \times (x-2) \) becomes \( 2x - 4 \).

By distributing and simplifying, we can quickly turn complex equations into manageable statements, making subsequent steps, like combining like terms, easier to apply.

Like Terms

Like terms in algebra are terms that have identical variable parts raised to the same power. Combining these makes expressions simpler and is an essential step in solving equations.

In the solution process of our exercise, after distributing, we have the expression \( 3x + 3 + 2x - 4 \). The like terms in this case are \( 3x \) and \( 2x \), which combine to give \( 5x \).

• Always group like terms together.
• Add or subtract their coefficients as needed.
• Simplify the expression to have fewer terms.

This simplification reduces the equation to a more straightforward form, such as \( 5x - 1 = 9 \), making it easier to isolate variables and find solutions.