Problem 49 - College Algebra

Question

contain rational equations with variables in denominators. For each equation, a. write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. $$ \frac{1}{x-4}-\frac{5}{x+2}-\frac{6}{x^{2}-2 x-8} $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 7\). The restrictions on \(x\) are \(4\) and \(-2\), and our solution is not a restriction, so we can confirm that it's valid.

Step 1: Find the restrictions

Finding the values of \(x\) that make the denominator zero are restrictions on the variable. Set each denominator equal to zero and solve for \(x\). We have \(x - 4 = 0\) and \(x + 2 = 0\) and \(x^{2} - 2x - 8 = 0\). Solving these yields restrictions on \(x\) as \(4, -2\) and \(4, -2\) from the quadratic equation.

Step 2: Simplify the equation

Observe that the expression in the third term \(\frac{6}{x^{2}-2 x-8}\) simplifies to \(\frac{6}{(x-4)(x+2)}\). The equation becomes \(\frac{1}{x-4}-\frac{5}{x+2}-\frac{6}{(x-4)(x+2)}\).

Step 3: Find a common denominator and combine terms

The least common denominator (LCD) is \((x-4)(x+2)\). Multiply each term by the LCD and simplify to combine the terms, which results in \(x+2 - 5(x-4) - 6 = 0\). This further simplifies to \(x- 5x+20 - 6 = -2x +14 = 0.\)

Step 4: Solve for x

Next, solve the equation \(-2x +14 = 0\) for \(x\). Adding \(2x\) to both sides yields \(14 = 2x\), and then dividing by 2 gives \(x=7\).

Step 5: Verify the solution

Having found the value of \(x\), it's now important to check if it's one of the restrictions. From Step 1 we have restrictions as \(4, -2\). Hence, the solution \(x = 7\) does not make the denominator zero, so it's a valid solution.

Key Concepts

Restrictions on Variables

Understanding the restrictions on variables is crucial when dealing with rational expressions and equations. A rational expression contains a numerator and a denominator, similar to a fraction, and variables can appear in either part. However, the expression is undefined when the denominator equals zero since division by zero is not permitted in mathematics. It is therefore necessary to determine which values, if any, would cause the denominator to be zero -- these values are the 'restrictions' on the variables.

To identify these restrictions, we set each denominator of the rational expression to zero and solve for the variable. For example, if we have denominators such as \(x-4\), \(x+2\), and \(x^2 - 2x - 8\), we equate each to zero and solve for \(x\). As a result, we determine that \(x\) cannot be 4 or -2. Keeping track of these restrictions is vital, as they are not part of the solution set and must be excluded from final answers.

In the context of educational online platforms, this concept is often visualized with number lines or by circling the restricted numbers on a graph, making it easier for students to recognize and comprehend the importance of restrictions on variables.

Finding the Common Denominator

When solving rational equations, the key step often involves finding a common denominator for all the terms. This process allows us to combine the fractions into a single expression or to clear the denominators to solve the equation more easily.

The common denominator must be a multiple of all the individual denominators in the equation. To find it, we typically factor each denominator (if not already in factored form), identify unique factors, and multiply them to form the least common denominator (LCD). For example, if the denominators are \(x-4\) and \(x+2\), the LCD would be their product \(x-4)(x+2)\). In educational resources, this concept is often highlighted through step-by-step factoring and multiplication, reinforcing in students the ability to recognize common factors and efficiently combine expressions.

Visualizing the LCD

Visual aides like Venn diagrams showing factors of each denominator can be used to help students identify the LCD. It's a way to make the abstract concept of common denominators more tangible, ultimately aiding in simplifying complex rational expressions.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that involves reducing expressions to their simplest form. Simplification can include various operations such as factoring, combining like terms, and canceling common factors in numerators and denominators. For rational expressions, simplification may involve dividing out common factors to reduce the fraction to lowest terms.

For instance, the expression \( \frac{6}{x^2 - 2x -8} \) can be simplified by factoring the quadratic equation in the denominator to \( \frac{6}{(x-4)(x+2)} \) and then further if there are common factors in the numerator.

Steps for Simplifying

Creating instructional content involves breaking down these steps and illustrating each one with examples. Diagrams that show factoring trees or use color-coding to highlight common factors can greatly enhance a student's understanding and retention of the method of simplifying algebraic expressions. By using these visual strategies, complex simplification processes can be made more accessible to learners.

Algebraic Solutions

Algebraic solutions refer to finding the value of variables that make the equation true. After simplifying the given rational equation and clearing the denominators, what remains is often a more straightforward linear or quadratic equation to solve.

In our example, the simplified form becomes \( -2x + 14 = 0 \)—a linear equation. To solve for \(x\), we perform algebraic operations such as adding, subtracting, multiplying, or dividing both sides of the equation by the same number to isolate \(x\) and find its value. In this instance, adding \(2x\) to both sides and then dividing by 2 yields the solution \(x=7\).

For educational content, demonstrating this process step by step ensures learners can follow and replicate the method. Additionally, presenting multiple examples with varying difficulty reinforces the concept and technique of solving for \(x\), underscoring the goal of mastering algebraic solutions.