Problem 20 - Contemporary Precalculus

Question

Use graphical approximation (a root finder or an intersection finder to find a solution of the equation in the given open interval. \(\frac{4}{x+2}-\frac{3}{x+1}=0 ; \quad(0, \infty)\) [Hint: Write the left side as a single fraction.]

Step-by-Step Solution

Answer

Answer: The solution is \(x = 2\).

Step 1: Simplify the equation

The equation we have is: \(\frac{4}{x+2}-\frac{3}{x+1}=0\) We should first simplify the left side of the equation. To do this, find a common denominator and combine the fractions. The common denominator of \(x+2\) and \(x+1\) is \((x+2)(x+1)\). Multiply both sides by the common denominator to eliminate the fractions: \((x+2)(x+1)\left(\frac{4}{x+2}-\frac{3}{x+1}\right) = 0\) Now, distribute and simplify: \(4(x+1) - 3(x+2) = 0\)

Step 2: Solve the simplified equation

Now we have: \(4(x+1) - 3(x+2) = 0\) Distribute the terms: \(4x + 4 - 3x - 6 = 0\) Combine like terms: \(x - 2 = 0\) Add 2 to both sides: \(x = 2\)

Step 3: Check if the solution is within the given interval

We found \(x = 2\). Now we need to check if this solution is within the given open interval (0, ∞). Since \(2 > 0\), the solution \(x = 2\) is within the interval, and thus it is a valid solution for the original equation.

Step 4: Final Answer

The solution for the equation \(\frac{4}{x+2}-\frac{3}{x+1}=0\) in the open interval \((0, \infty)\) is \(x = 2\).

Key Concepts

Graphical Approximation

When solving equations, especially in calculus or higher-level math courses, graphical approximation is a valuable problem-solving strategy. It involves visually estimating the solution of an equation using a graph. This approach can give you a good idea of where the solution lies, even before you perform any algebraic manipulation.

Using a graphing calculator or computer software, you can plot each side of the equation as separate functions and look for the points of intersection. These intersections represent the x-values where both sides of the equation equal each other. In the given exercise, plotting \(\frac{4}{x+2}\) and \(\frac{3}{x+1}\) on a graph and finding their intersection within the interval \(0, \infty)\) would approximate the solution to the equation.

Common Denominator

Dealing with equations involving fractions, a 'common denominator' is the key to combining and simplifying those fractions. This term refers to a shared multiple of the denominators. By finding the least common denominator (LCD), you can add or subtract fractions much more easily because they become compatible.

In the case of our rational equation, the denominators were \(x+2\) and \(x+1\). Their LCD is the product of the two, \(x+2)(x+1)\), as neither denominator is a multiple of the other. Multiplying the entire equation by this common denominator eliminates the fractions, which simplifies the equation and brings us closer to a solution.

Interval Solution

An interval solution is an answer to an equation that is restricted to a specific range of values, known as an interval. For our rational equation, the interval given was \(0, \infty)\), which means that we're only interested in solutions where \(x\) is greater than zero.

To verify our solution is valid in this context, we check whether the value of \(x\) we derived from simplifying the equation falls within that interval. In our exercise, our answer was \(x=2\), which indeed lies within the interval \(0, \infty)\). Thus, \(x=2\) is an interval solution.

Simplify Equations

Simplifying equations is fundamental to solving math problems effectively. This process can involve combining like terms, canceling elements, and reducing fractions to lower terms. The goal is to make the equation as straightforward as possible, often narrowing it down to something that's easily solvable algebraically.

In our exercise, we simplified the original equation by finding a common denominator and distributing terms, which brought us to a linear equation with a clear solution. This step-by-step simplification was crucial in finding the valid solution of \(x=2\) for the rational equation given.