# solving rational equations

So x+3 and (x+3)/1 both have the same value, but the latter expression is considered a rational expression, because it's written as a fraction. When we solve rational equations, we can multiply both sides of the equations by the least common denominator (which is $$\displaystyle \frac{{\text{least common denominator}}}{1}$$ in fraction form) and not even worry about working with fractions! Let's begin by looking at solving an equation with rational functions in it. Factor 3 out of both numerator and denominator. Unfortunately, this method only works for rational equations that contain exactly one rational expression or fraction on each side of the equals sign. If you really can’t stand to see another ad again, then please consider supporting our work with a contribution to wikiHow. How do I solve this rational equation? If we wish, this can also be written as -2x - 6 = 4x. That is, all I really need to do now is solve the numerators: Since x = 4/3 won't cause any division-by-zero problems in the fractions in the original equation, then this solution is valid. If you got the variable value correct, you'll be able to simplify the original equation to a simple valid statement, such as 1 = 1. This article has been viewed 99,758 times. Solving rational equations with variables in the denominators involves manipulating and rewriting the terms. Rational expressions and other fractions can be made into non-fractions by multiplying them by their denominators. To solve a rational equation, start by rearranging it so you have 1 fraction on each side of the equals sign. So x = +/- 8. Subtract 1 from both sides to get 2x+2 = 3x, and subtract 2x from both sides to get 2 = x, which can be written as x = 2. I'll show each, and you can pick whichever you prefer. These "new forms" of the original equation may produce solutions that do not work in the original equation. Multiply both sides by 6 to cancel the denominators, which leaves us with 2x+3 = 3x+1. These fractions may be on one or both sides of the equation. Learn to use Zoom in this beginner-friendly course. In our example, we can divide both sides of the equation by -2, giving us x+3 = -2x. 1/3(2m-12) expands to 1/(6m-36). Q (x) P (x) . Multiply both sides of the equation by (x-3). Please help us continue to provide you with our trusted how-to guides and videos for free by whitelisting wikiHow on your ad blocker. Purplemath. Solving Equations Video Lesson. In this section, we look at rational equations that, after some manipulation, result in a linear equation. Recall that you can solve equations containing fractions by using the least common denominator of all the fractions in the equation. Since this is an equation, I can multiply through by whatever I like. An equation involving rational expressions is called a rational equation. Multiply both sides of the equation by 13. A common way to solve these equations is to reduce the fractions to a common denominator and then solve the equality of the numerators. For example, in the equation x/8 + 2/6 = (x - 3)/9, the LCD is 8*9 = 72. Extraneous solutions are solutions that don't satisfy the original form of the equation because they produce untrue statements or are excluded values that make a denominator equal to 0. Also, there's the new wrinkle of variables in the denominator. To illustrate this, let’s look at a very simple equation: x = 3 . Cross-multiplication is basically a handy shortcut for multiplying both sides of the equation by both fraction's denominators. You should use the method that works best for you. This works to 15x = 3x - 3 + 2x -2, which simplifies to 15x = x - 5. To create this article, 12 people, some anonymous, worked to edit and improve it over time. These "new forms" of the original equation may produce solutions that do not work in the original equation. Solving Rational Equations Rational equations are simply equations with rational expressions in them. After clearing the fractions, we will be left with either a linear or a quadratic equation that can be solved as usual. Examples are 1 (which is 1/1), 34 (which is 34/1), 2/3, and 48/37. Like normal algebraic equations, rational equations are solved by performing the same operations to both sides of the equation until the variable is isolated on one side of the equals sign. If an equation contains at least one rational expression, it is a considered a rational equation.. Recall that a rational number is the ratio of two numbers, such as $\frac{2}{3}$ or $\frac{7}{2}$. If either side of the equation has added (or subtracted) fractions, we must use Method 1 or Method 2. This is because, as soon as you go from a rational expression (that is, something with no "equals" sign in it) to a rational equation (that is, something with an "equals" sign in the middle), you get a whole different set of tools to work with.

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