Problem 26 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the equation. $$\frac{9}{2 x+6}-\frac{7}{5 x+15}=\frac{2}{3}$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{33}{20}\) or 1.65.

Step 1: Simplify the Denominators

Both denominators, \(2x + 6\) and \(5x + 15\), can be simplified by factoring out common factors.\[2x + 6 = 2(x + 3)\] and \[5x + 15 = 5(x + 3)\].

Step 2: Find a Common Denominator

The common denominator for the fractions \(\frac{9}{2(x+3)}\) and \(\frac{7}{5(x+3)}\) is \(10(x+3)\) because this is the LCM of \(2(x+3)\) and \(5(x+3)\).

Step 3: Combine the Fractions

Rewrite each fraction with the common denominator: \[\frac{9}{2(x+3)}=\frac{45}{10(x+3)}\] and \[\frac{7}{5(x+3)}=\frac{14}{10(x+3)}\]. The equation becomes:\[\frac{45}{10(x+3)}-\frac{14}{10(x+3)}=\frac{2}{3}\].

Step 4: Clear the Fractions

Combine the terms on the left: \[\frac{31}{10(x+3)}=\frac{2}{3}\]. Next, eliminate the fractions by cross-multiplying:\[31 \cdot 3 = 2 \cdot 10(x+3)\].

Step 5: Solve for x

Simplify and solve the equation:\[93 = 20(x+3)\] Divide both sides by 20: \[x+3 = \frac{93}{20}\]Subtract 3 (or \(\frac{60}{20}\)) from both sides:\[x = \frac{93}{20} - \frac{60}{20}\] = \[\frac{33}{20}\] or 1.65.

Key Concepts

Equation Solving

Solving equations is a critical skill in algebra, allowing you to find the value of unknown variables. An equation is a mathematical statement where two expressions are equal. To solve an equation like \( \frac{9}{2x+6} - \frac{7}{5x+15} = \frac{2}{3} \), you primarily aim to isolate the variable \( x \) on one side. This process requires steps such as simplifying expressions, finding common denominators, and performing arithmetic operations like addition or multiplication on both sides. Breaking down each operation ensures that the balance of the equation is maintained, ultimately leading to discovering the value that satisfies the original equation.

Rational Equations

Rational equations involve fractions where the numerator and/or the denominator are polynomials. In our exercise, the rational equation we're dealing with involves variables in the denominators. Key points to remember include:

• Always look for ways to simplify the equation by factoring polynomials.
• The objective is to eliminate the fractions by finding a common denominator or through multiplication. 
• Cross-multiplication is often used as a method to clear fractions when both sides of the equation are fractions.

Rational equations can seem complex due to the presence of fractions, but with careful simplification and strategic operations, they can be solved efficiently.

Factoring

Factoring is an essential algebraic tool used to break down expressions into simpler components. It is particularly useful when dealing with polynomials in rational equations, as seen with our denominators \( 2x + 6 \) and \( 5x + 15 \). Factoring involves finding common factors:

• The expression \( 2x + 6 \) can be factored as \( 2(x+3) \).
• Similarly, \( 5x + 15 \) becomes \( 5(x+3) \).

By identifying these common factors, you can simplify expressions and make it easier to find common denominators, ultimately facilitating the solution of the equation. Remember, efficient factoring can greatly reduce the complexity of an algebraic problem.

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions. It allows the fractions to be combined or compared directly, which is essential for solving rational equations like our exercise. To find a common denominator, you need to determine the least common multiple (LCM) of the denominators. For \( 2(x+3) \) and \( 5(x+3) \), the common denominator is \( 10(x+3) \). Steps include:

• Write each fraction with the same denominator by adjusting the numerators accordingly.
• Once the fractions share a common denominator, they can be combined for easier solving.

Having a common denominator simplifies subtraction or addition in equations, allowing you to focus on isolating the variable to solve the equation effectively.