Problem 68 - College Algebra

Question

The equations in Exercises \(59-70\) combine the types of equations we have discussed in this section. Solve each equation or state that it is true for all real numbers or no real numbers. $$ \frac{1}{x-1}=\frac{1}{(2 x+3)(x-1)}+\frac{4}{2 x+3} $$

Step-by-Step Solution

Answer

The solution for x is \(x = \frac{1}{3}\)

Step 1: Identify the LCD

The Least Common Denominator (LCD) is \( (2x+3)(x-1) \) as it covers all the denominators in the equation.

Step 2: Multiply by the LCD

Multiply each term of the equation with the LCD: \( (2x+3)(x-1) \cdot \frac{1}{x-1} = (2x+3)(x-1) \cdot \frac{1}{(2x+3)(x-1)} + (2x+3)(x-1) \cdot \frac{4}{2x+3} \). This gets rid of the fractions in the equation.

Step 3: Simplify

Simplify the equation: \( (2x+3) - 1 + 4(x-1) \). This simplifies to: \(2x + 3 - 1 + 4x - 4 \), which gives: \(6x - 2\).

Step 4: Finding x

As per the solution after simplification, the equation becomes \(6x - 2 = 0\). Solving this for x gives: \(6x = 2 \), so \(x = \frac{2}{6}\), which simplifies to \(x = \frac{1}{3}\)

Key Concepts

least common denominator

When dealing with rational equations, you'll often encounter terms with different denominators. To simplify solving these equations, finding the Least Common Denominator (LCD) is key. The LCD is essentially the smallest expression that all the denominators in the equation can divide into without a remainder.

For example, in our original exercise, the denominators were \((x-1)\) and \((2x+3)\), so the LCD is \((2x+3)(x-1)\). The reason we find the LCD is simple: it helps us eliminate the fractions by allowing us to multiply each term in the equation by this common denominator, simplifying the equation to a polynomial without fractions.

• Locate all denominators in the equation.
• Multiply each denominator to find the LCD.
• The LCD should be distributed across each term to clear away the fractions.

rational equations

Rational equations involve variables within the denominators of fractions. These types of equations can sometimes appear daunting because they involve fractions and potential restrictions on the variables due to undefined expressions at certain points.

To solve rational equations effectively, the first step is identifying and using the LCD. This principal task transforms the equation, enabling manipulation similar to regular linear or polynomial equations without the complication of fractions.

When working with rational equations, remember:

• Ensure denominators don't equal zero, as this makes the expression undefined.
• The steps involve cross-multiplying by the LCD to clear fractions.
• This converts the equation into a solvable polynomial form.

Using these methods, you can navigate and solve rational equations with ease, ensuring to always check for extraneous solutions that might arise after clearing the denominators.

solving equations

Solving equations is a fundamental process in algebra used to find the value of the variable that makes the equation true. After using the LCD to remove fractions in rational equations, you're left with a simpler equation that can be solved using algebraic principles.

Let's break down the solving process that was demonstrated in our solution:

• Initially, the equation transforms into a simpler polynomial after multiplying by the LCD.
• Simplify by combining like terms, as seen when \(2x + 3 - 1 + 4x - 4\) reduces to \(6x - 2\).
• Set the equation equal to zero and isolate \(x\).
• Solve for \(x\) by isolating it on one side of the equation: \(6x = 2\) becomes \(x = \frac{1}{3}\).

Always verify your solutions by plugging them back into the original equation to ensure they don't make any denominator zero, thus validating the legitimacy of your answer.