Problem 34 - 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. $$\frac{7}{2 x}-\frac{5}{3 x}=\frac{22}{3}$$

Step-by-Step Solution

Answer

The restrictions on the variable x is x≠0, meaning x cannot be equal to zero. And solving the equation while keeping this restriction in mind gives the answer: \(x=0.25\).

Step 1: Identify the Restrictions

The restrictions on the variable x are the values that make the denominator zero. Looking at denominators in our problem, 2x and 3x, these would be zero when x=0. Therefore, the restriction on variable x is that x≠0.

Step 2: Simplifying the Equation

To simplify this type of equation, it is easier to find a common denominator and use it to eliminate fractions. The common denominator of 2x and 3x is 6x. Multiply each side of the equation by the common denominator: \(6x*(\frac{7}{2x} - \frac{5}{3x}) = 6x*\frac{22}{3}\). After simplifying, the equation becomes \(21 - 10 = 44x\).

Step 3: Solve for x

Further simplifying step 2, we get \(11 = 44x\), dividing both sides by 44, we get \(x = \frac{11}{44} = 0.25\). Remembering our restrictions, x=0.25 is a valid solution because it does not make the denominator zero.

Key Concepts

Rational Equations Restrictions

When working with rational equations, one crucial step that we must never skip is identifying the restrictions on the variables. These restrictions arise from the need to prevent the denominators from equating to zero, as division by zero is undefined in mathematics.

For instance, consider the rational equation \( \frac{7}{2x} - \frac{5}{3x} = \frac{22}{3} \). The denominators here, 2x and 3x, would both become zero if x were equal to zero. Hence, the restriction for this equation is that x must not be equal to zero, which we express mathematically as x≠0. This is a non-negotiable rule because, otherwise, the expression becomes meaningless within the realm of real numbers.

Always begin by identifying these restrictions and recording them separately from your solution process. This ensures that any solutions you find later do not violate the conditions that define the domain of valid answers.

Common Denominator in Equations

To streamline the process of solving rational equations, finding a common denominator is a powerful technique. A common denominator allows us to combine the terms which have different denominators and simplifies the equation into a simpler form that we can solve more easily.

In our example \( \frac{7}{2x} - \frac{5}{3x} = \frac{22}{3} \), the least common denominator (LCD) for the denominators 2x and 3x is 6x. By multiplying every term in the equation by this LCD, we can eliminate the fractions, resulting in an equation without denominators: \(6x*(\frac{7}{2x}) - 6x*(\frac{5}{3x}) = 6x*\frac{22}{3}\), which simplifies to \(21 - 10 = 44x\).

Finding the common denominator not only simplifies our calculations but also helps us see the structure of the equation more clearly, leading to quicker and more accurate solutions.

Solving for Variables

After addressing restrictions and common denominators, the next step in tackling a rational equation is to solve for the variable. This involves isolating the variable on one side of the equation to determine its value. Uses of operations such as addition, subtraction, multiplication, or division are key to achieving this goal.

Continuing from our earlier steps in the problem \( \frac{7}{2x} - \frac{5}{3x} = \frac{22}{3} \), after finding the common denominator and simplifying, we obtained \(11 = 44x\). Now, to isolate x, divide both sides of the equation by 44, yielding \(x = \frac{11}{44} = 0.25\).

Ensure that your final answer does not conflict with any earlier stated restrictions. In our case, x = 0.25 is an acceptable solution because it complies with the established restriction x≠0. This final check is important, as it confirms the validity of your solution within the defined parameters of the equation.