Problem 13 - Elementary Algebra

Question

Solve each of the equations. $$\frac{x+1}{3}-\frac{x+2}{2}=4$$

Step-by-Step Solution

Answer

x = -28

Step 1: Identify the terms

Examine the equation \( \frac{x+1}{3}-\frac{x+2}{2}=4 \). We have two fractions on the left side with different denominators.

Step 2: Find a common denominator

The denominators are 3 and 2. The least common denominator (LCD) is 6. Rewrite the fractions to have this common denominator.

Step 3: Rewrite the fractions

Convert \( \frac{x+1}{3} \) to \( \frac{2(x+1)}{6} \) and \( \frac{x+2}{2} \) to \( \frac{3(x+2)}{6} \). The equation is now \( \frac{2(x+1)}{6} - \frac{3(x+2)}{6} = 4 \).

Step 4: Combine the fractions

Since both fractions have the same denominator, combine them: \[ \frac{2(x+1) - 3(x+2)}{6} = 4 \].

Step 5: Simplify the numerator

Distribute within the numerator: \[ 2(x+1) = 2x + 2 \] and \[ 3(x+2) = 3x + 6 \]. Then, simplify: \[ 2x + 2 - 3x - 6 = -x - 4 \].

Step 6: Clear the fraction

To eliminate the fraction, multiply both sides of the equation by 6: \[ -x - 4 = 24 \].

Step 7: Solve for x

Add 4 to both sides: \[ -x = 28 \]. Then multiply by -1: \[ x = -28 \].

Key Concepts

Fractions in Equations

When working with equations that contain fractions, it's crucial to manage and manipulate the fractional components effectively. Fractions have numerators and denominators, which can make equations look more complicated than they are.
To simplify fractions in equations and make calculations easier to manage, follow these steps:

• Identify the fractional terms in the equation.
• Check if the fractions have common denominators. This is important for combining or simplifying fractions later on.
• If needed, rewrite fractions to have a common denominator. This step is essential for correctly adding or subtracting fractions.

Taking the time to carefully handle fractions can make solving linear equations much smoother. Being attentive in these steps prevents errors and misunderstandings that can arise from fractional terms.

Least Common Denominator

Finding the least common denominator (LCD) is a critical step in solving equations with fractions. The least common denominator is the smallest number that can be a common denominator for all fractions involved. Here's how you can find it:

• Identify all the denominators in your equation. These are the numbers at the bottom of each fraction.
• Find the smallest number that all the denominators divide into. This is your least common denominator.
• Rewrite each fraction with this common denominator by adjusting the numerators accordingly. This means multiplying the numerator by whatever factor you needed to multiply the original denominator to get the LCD.

This process helps to efficiently manage fractional terms, ensuring they are ready for further calculation. For instance, in the provided exercise, the denominators 3 and 2 lead to an LCD of 6. Converting fractions to have this denominator allows for simple subtraction and combination.

Solving Equations

Once fractions have been managed and a common denominator applied, solving the equation becomes more straightforward. Solving linear equations typically involves isolating the variable, such as \( x \), as your main goal. Here's a basic approach:

• Combine or simplify any terms that you can. This includes removing any common denominators from fractions.
• Convert the equation so the variable is isolated on one side. This may involve adding, subtracting, multiplying, or dividing both sides of the equation.
• Solve for the variable by performing the necessary operations until \( x \) or another variable stands alone.

In our example, after simplifying with the LCD, we clear fractions by multiplying through by the denominator and then isolate \( x \). The operation yields a solution for \( x = -28 \), solved step-by-step to ensure clarity and accuracy.