Problem 36 - 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. $$ \frac{5}{2 x}-\frac{8}{9}=\frac{1}{18}-\frac{1}{3 x} $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 0.2941\). The value \(x = 0\) is a restriction and is not a valid solution for this equation.

Step 1: Identify the Restrictions

To identify the restrictions, we need to set the denominators equal to zero and solve for \(x\). The denominators here are \(2x\) and \(3x\). Setting \(2x = 0\) and \(3x = 0\), we get \(x = 0\) as restriction.

Step 2: Clear the Fractions

Now, to solve the equation, we will first get rid of the denominators by multiplying each term on both sides by the Lowest Common Denominator (LCD) which is \(18x\). The equation becomes: \(45x^2-16x= x-6x^2.\)

Step 3: Simplify and Solve the Equation

Next, we'll combine like terms and bring the terms to one side of the equation, leading to: \(51x^2-16x +x=0.\) This simplifies to: \(51x^2-15x=0.\) Then we factor by \(x\) to solve for \(x\), ending up with:\(x(51x-15)=0.\) Setting each factor equal to zero gives two solutions \(x = 0\) and \(x = 15/51 = 0.2941\).

Step 4: Check the Restrictions

Recall from step 1 that the restriction is \(x≠0\), therefore, \(x = 0\) is not a solution. Hence, \(x = 0.2941\) is the only valid solution for this equation.

Key Concepts

Identifying Restrictions in Rational Equations

Identifying restrictions in rational equations is an essential first step to ensure that the solution to the equation is valid. These restrictions come from the fact that division by zero is undefined in mathematics. In any rational equation, if you have a variable in the denominator, you must consider what value of this variable would make the denominator zero.

For instance, let’s consider the rational equation \(\frac{5}{2x}-\frac{8}{9}=\frac{1}{18}-\frac{1}{3x}\). The denominators here are \(2x\) and \(3x\), and setting them equal to zero gives us the restrictions. We solve \(2x = 0\) and \(3x = 0\) to find that \(x = 0\) is the value that would make these denominators zero. Thus, \(x ≠ 0\) is our restriction for this equation.

Finding Least Common Denominator (LCD)

When solving equations with fractions, finding the Least Common Denominator (LCD) is a crucial step to create equivalent fractions and eliminate the fractions from the equation. The LCD is the smallest number that each of the denominators can divide into without leaving a remainder.

In our exercise, the denominators are 2, 18, and 9, when considering the variable part \(x\), the LCD must include it as well. Multiplying these values together we might get a common denominator, but it's not the least. In fact, for our denominators, the LCD is \(18x\), since 18 is the LCM of 2, 9, and 18, and we include the variable \(x\) to account for \(2x\) and \(3x\). By finding the LCD, we can proficiently move forward to clear the fractions and solve the equation.

Clearing Fractions in Equations

Clearing fractions in an equation simplifies the process of solving it by transforming it into an equation without fractions, which is typically easier to solve. To clear the fractions, we multiply every term on both sides of the equation by the LCD that we’ve found.

In our example, we multiply each term by the LCD, \(18x\), resulting in \(45x^2 - 16x = x - 6x^2\). This step effectively clears the fractions and sets the stage for combining like terms and moving all terms to one side of the equation. It is equivalent to finding a common base for each term, further simplifying the equation solving process.

Factoring Algebraic Expressions

Factoring is a method used to simplify algebraic expressions and to solve equations in a more manageable form. By factoring, we express an expression as a product of its factors, which are simpler expressions.

In the context of our equation, once we cleared the fractions and combined like terms, we got \(51x^2 - 15x = 0\). To factor this expression, we look for common factors in each term. Here, \(x\) is a common factor, and we can write the equation as \(x(51x - 15) = 0\). This reveals the solutions to the equation because the product of factors is zero when one or more of the factors are zero. It allows us to set each factor equal to zero and solve for \(x\), ensuring that the solutions adhere to the restrictions previously identified.