Problem 40 - 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. $$ \frac{3}{x+4}-7-\frac{-4}{x+4} $$

Step-by-Step Solution

Answer

The restriction on the variable \(x\) is \(x = -4\). Keeping this restriction in mind, the solution to the equation is \(x = -29/7.\)

Step 1: Find Restrictions on Variable

To find restrictions on the variable, set the denominator to zero and solve for \(x\): \[x+4=0\]\nSolving for \(x\) gives \(x = -4\). So, \(x = -4\) is the restriction for this equation.

Step 2: Solve the Rational Equation

We can now solve the equation while keeping in mind that \( x \neq -4 \): \[\frac{3}{x+4}-7- \frac{-4}{x+4} = 0\]\nThis simplifies to \[\frac{3 - 4}{x+4} - 7 = 0\]. Now we can combine the two fractions by finding a common denominator, which in this case is \(x+4\), giving:\[\frac{-1}{x+4} - 7 = 0.\]\nAdd 7 to both sides and simplify to obtain: \[\frac{-1}{x+4} = 7.\]\nNow, cross multiply to solve for \(x\):\[-1 = 7(x+4).\] This simplifies to \[-1 = 7x + 28,\] where you subtract 28 from both sides to give \[-29 = 7x.\] So, \(x = -29/7.\)

Step 3: Check the Solution Against the Restrictions

The solution found is \(x = -29/7,\) which doesn't equal to the restriction \(x = -4.\) Therefore, the solution is valid.

Key Concepts

Restrictions on Variables

In rational equations, variables often appear in denominators, and it's crucial to determine if any part of the equation makes these denominators zero. This is what we call checking for restrictions on the variables. If a variable causes a division by zero, it is not allowed, as division by zero is undefined.
To find restrictions, take each denominator in your equation and set it equal to zero. Solving these equations gives you the values that are not permissible for the variable.

• For the exercise given: We set the denominator \(x+4\) equal to zero.
• Solving, we find that \(x = -4\).

This means \(x = -4\) is a restriction and should not be a part of any viable solution for the equation. Always remember this step before proceeding with solving rational equations.

Solving Rational Equations

Once you've identified restrictions, it’s time to solve the rational equation. This involves finding a value for the variable that satisfies the equation while still considering the restrictions found earlier.
For this, substitute the expressions as needed and then simplify:

• Our example equation is \(\frac{3}{x+4} - 7 - \frac{-4}{x+4} = 0\).
• Combine like terms; here, combining \(\frac{3}{x+4}\) and \(\frac{-4}{x+4}\) simplifies to \(\frac{-1}{x+4}\).
• Continue simplifying by isolating the variable term.

It's important to manipulate the equation carefully to solve for \(x\) while never making \(x = -4 \).

Common Denominator Method

The Common Denominator Method is a straightforward approach to handle rational equations effectively, especially when dealing with multiple fractions.
This method requires:

• Identifying a common denominator for all fractions involved in the equation. In our exercise, \(x+4\) is the common denominator.
• Rewriting each term of the equation with this common denominator, allowing you to easily combine the fractions.
• Our example shows this with \(\frac{-1}{x+4} - 7 = 0\).

After combining, solving involves isolating the term with the variable, followed by cross-multiplying if needed.
The ultimate goal is to arrive at a workable form of the equation where solving for \(x\) becomes clear and manageable, without breaking the restriction identified. In this case, the final solution, \(x = -\frac{29}{7}\), fits as \(x\) is not \(-4\).