Problem 40 - College Algebra

Question

Exercises \(31-50\) contain 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. $$ \frac{3}{x+4}-7=\frac{-4}{x+4} $$

Step-by-Step Solution

Answer

The restriction on the variable x for the given rational equation is that \(x \neq -4\). After simplifying and solving the equation, we find that the solution is \(x = 1\).

Step 1: Identify Restrictions

To identify restrictions, we need to find the values of x that will make the denominator equal zero. In other words, we solve \(x + 4 = 0\). After rearranging, we get \(x = -4\). Therefore, the restriction on the variable x is that it cannot equal -4.

Step 2: Simplify the Equation

Keeping the restriction in mind, we proceed to solve the equation. Since the denominators on both sides of the equation are the same, we can rewrite the equation by setting the numerators equal: \(3 - 7x = -4\). This simplifies to \(7x = 3 + 4\) or \(7x = 7\).

Step 3: Solve for x

Now that we have a simplified linear equation, we can solve it for x. We divide both sides by 7: \(x = 7 / 7 = 1\). The root of the equation is 1, but it must be different from the restriction we found in Step 1.

Key Concepts

Identifying Restrictions in Rational Equations

When faced with rational equations, where variables appear in the denominators, the first step is identifying restrictions—values that the variable cannot take. For instance, consider the equation \(\frac{3}{x+4}-7=\frac{-4}{x+4}\). Here, \(x+4\) is a denominator that contains a variable which can potentially become zero. And as we may recall, division by zero is undefined in mathematics, implying that any value making the denominator zero is not allowed. To identify the restrictions, we set the denominator equal to zero and solve for the variable: \(x+4=0\), yielding \(x=-4\). This outcome is a restriction, meaning that \(x\) cannot equal -4. Understanding and pinpointing these restrictions are crucial before attempting to solve the equation, as these values must be excluded from the final solution set.

Solving Rational Equations

Solving rational equations requires manipulation to isolate the variable on one side of the equation while keeping in mind the restrictions identified earlier. In the given exercise, we started with \(\frac{3}{x+4}-7=\frac{-4}{x+4}\). Since the denominators are identical, we can set the numerators equal to each other: \(3 - 7(x + 4) = -4\). After distributing and combining like terms, the equation is simplified to a linear one which can be solved in a straightforward manner. Clearing the fractions leads us to \(7x = 7\), and dividing both sides by 7 gives \(x = 1\). It's vital to simplify the equation step by step, eliminating fractions when possible, to uncover a linear equation that can be tackled with elementary algebraic techniques.

Finding Denominators Equal to Zero

In rational equations, finding denominators equal to zero is a safeguard against undefined expressions. Expressions like \(\frac{3}{x+4}\) have the potential hazard of an undefined result if the denominator equals zero. To avoid this, we determine the values that would cause such a scenario before solving the equation. From \(x+4=0\), we find that \(x=-4\) is the problematic value. By doing this, we make sure that our subsequent solution does not include these undefined points, ensuring the mathematical validity of our answers. Denominators equal to zero are the 'off-limits' in the world of rational expressions, and identifying them is an essential safety check in the solution process.