Problem 40 - Algebra and Trigonometry

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. $$\frac{3}{x+4}-7=\frac{-4}{x+4}$$

Step-by-Step Solution

Answer

The restriction on the variable 'x' is that it cannot equal -4. The solution for the variable 'x' that satisfies the given equation is \(x = 2\).

Step 1: Identify the restrictions

Set the denominator to zero to find the values that would make it zero. This will yield the restrictions. Given that the denominator is \(x+4\), we find the restriction as follows: \(x+4=0\) which further simplifies to \(x=-4\). Thus the restriction on the variable \(x\) is that it cannot be equal to -4.

Step 2: Solving the rational equation

Keeping the restriction in mind (\(x \neq -4\)), solve the given equation \(\frac{3}{x+4}-7=\frac{-4}{x+4}\). We observe that the denominators are the same, therefore we can equate the numerators. This simplifies to \(3-7(x+4)=-4\) . Solving this linear equation, we get \(x = 2\).

Step 3: Check the solution

Though \(x = 2\) satisfies the equation, we need to ensure it doesn't violate the restriction established in step 1. Since \(2 \neq -4\), this is a valid solution.

Key Concepts

Restrictions on Variables

When dealing with rational equations, especially those that have variables in the denominators, it's crucial to determine the restrictions on the variables. This step ensures the integrity of the equation by identifying values that the variable cannot take, as these would lead to division by zero, which is undefined in mathematics.

In the given equation \( \frac{3}{x+4} - 7 = \frac{-4}{x+4} \), we first focus on the denominator, which is \( x + 4 \). To find the restriction, we set the denominator equal to zero:

• \(x + 4 = 0\)
• Solve for \(x\): \(x = -4\)

This indicates that \(x\) cannot be \(-4\). Any value for \(x\) other than \(-4\) keeps the equation valid and defined.

Solving Equations

Solving rational equations follows similar steps to solving other equations, but with additional attention to any restrictions. Once we've accounted for restrictions, it's possible to manipulate and simplify the equation to find valid solutions.

In our rational equation \( \frac{3}{x+4} - 7 = \frac{-4}{x+4} \), the denominators are identical, which simplifies our task. Instead of dealing with complex expressions, we can set the numerators equal to each other:

• Equation becomes \( 3 - 7 = -4 \)
• This is simplified to \( 3 - 7(x + 4) = -4 \)

Solving this linear equation results in \(x = 2\). It is crucial to verify that this solution does not conflict with our earlier restrictions.

Denominators in Equations

Denominators in rational equations can introduce special challenges, as they determine the restrictions on possible solutions. They also require careful planning when manipulating equations as they are part of the fraction expression.

In our example, the equation \( \frac{3}{x+4} - 7 = \frac{-4}{x+4} \) continued the constant denominator \(x + 4\). Since this term is shared between rational expressions, it allows straightforward simplification by focusing on the numerators.

• Common denominators simplify solving since numerators can be directly equated.
• Replacing the denominator's variable with restricted values leads to undefined expressions, which are avoided by applying restrictions effectively.

Proper handling of denominators ensures correct solutions while maintaining the overall validity of mathematical equations.