Problem 21 - Introductory Algebra for College Students

Question

Solve each rational equation. $$x+\frac{3}{x}=\frac{12}{x}$$

Step-by-Step Solution

Answer

The solution to the equation \(x + \frac{3}{x} = \frac{12}{x}\) is \(x = 3, -3\).

Step 1: Simplification

The first step involves simplifying the equation and putting it in the standard form. In this case, the equation \(x + \frac{3}{x} = \frac{12}{x}\) can be rewritten as \(x = \frac{12}{x} - \frac{3}{x}\). Now we are ready to move forward to the next step.

Step 2: Clear fractions

We will clear the fraction by finding the least common multiple (LCM) of the denominators. The denominators are \(x\) in both terms. Therefore, by multiplying every term by \(x\), we get \(x^2 = 12 - 3\).

Step 3: Solve for x

Now we can simplify and solve for x. It will involve solving the equation \(x^2 = 9\). The solution to this equation is \(x = \pm3\) (since both 3 and -3 squared provides 9).

Step 4: Check solutions

Finally, we need to substitute \(x = 3\) and \(x = -3\) back into the original equation to ensure the solutions are valid. We must not end up with a zero in any denominator. When we do this, we find that both values are valid solutions.

Key Concepts

Algebraic Fractions

When tackling algebra problems, you will often come across algebraic fractions. These are expressions that feature a polynomial in the numerator, the denominator, or both. The challenge with algebraic fractions lies in performing operations such as addition, subtraction, multiplication, and division, much like you do with regular numbers.

To solve equations with algebraic fractions, the goal is to eliminate the fractions and simplify the equation. One effective method is to find a common denominator and multiply through to clear the fractions. In our exercise, we cleared the fraction by multiplying every term by the denominator, which is a common tactic used to simplify the equation into a more workable form. After clearing the fractions, it's just a matter of simplifying and solving for the unknown variable.

Least Common Multiple

Least common multiple (LCM) is a concept that is frequently employed when working with algebraic fractions. LCM is the smallest number that is a multiple of two or more numbers. For instance, the LCM of 4 and 6 is 12, because it is the smallest number that both 4 and 6 can divide into without leaving a remainder.

Finding the LCM is crucial when you want to combine fractions or clear fractions from an equation. By multiplying each term by the LCM, you ensure all fractions are eliminated. This step simplifies the equation into something that looks far less intimidating and much closer to a standard polynomial equation. Keeping an eye out for the LCM is a handy skill, as it can make complex-looking equations far easier to solve.

Quadratic Equations

A quadratic equation is an equation of the second degree, meaning it contains at least one term that is squared. The general form of a quadratic equation is \(ax^2 + bx + c = 0\). These equations can be solved through factoring, completing the square, using the quadratic formula, or graphing. In the exercise provided, after clearing the algebraic fraction, we arrived at the simplified equation \(x^2 = 9\), which is a basic quadratic equation.

To solve this equation, we looked for values of \(x\) that satisfy the equation when squared. The solutions, \(x = +3\) and \(x = -3\), are derived from the fact that both \(3^2\) and \( (-3)^2\) equal 9. It's imperative to always check that the solutions found do not create any undefined conditions in the original equation, such as division by zero.