Solve each rational equation. $$\frac{3}{x+4}+\frac{7}{x-4}=\frac{8}{x^{2}-16}$$
Short Answer
Expert verified
x = -\frac{4}{5}
Step by step solution
01
Identify the denominators
The denominators in the given rational equation are \(x + 4\), \(x - 4\), and \(x^{2} - 16\). Notice that \(x^{2} - 16\) can be factored as \( (x + 4)(x - 4)\).
02
Find a common denominator
The common denominator for the equation is \( (x + 4)(x - 4)\) since \(x^{2} - 16 = (x + 4)(x - 4)\).
03
Multiply each term by the common denominator
Multiply each term in the equation by \( (x + 4)(x - 4) \) to clear the denominators. You get: \[ 3(x - 4) + 7(x + 4) = 8 \]
04
Simplify the equation
Distribute and combine like terms: \[ 3x - 12 + 7x + 28 = 8 \] Combine the \(x\) terms and constants: \[ 10x + 16 = 8 \]
05
Solve for x
Isolate \(x\) by first subtracting 16 from both sides: \[ 10x = -8 \] Then, divide both sides by 10: \[ x = -\frac{8}{10} = -\frac{4}{5} \]
06
Check for extraneous solutions
Verify whether \( x = -\frac{4}{5} \) is a valid solution by plugging it back in the original equation and confirming if both sides are equal. Since \(x = -\frac{4}{5} \) does not make any denominator zero, it is a valid solution.
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Factoring Polynomials
Factoring polynomials is an essential skill when working with rational equations. It involves expressing a polynomial as a product of simpler polynomials. For example, consider the polynomial expression in the denominator of our rational equation: \(x^{2} - 16\). This can be factored into \((x + 4)(x - 4)\), using the difference of squares formula. Factoring helps in simplifying complex rational expressions and finding common denominators. When you successfully factor polynomials, it often leads to easier calculations and clearer solutions in rational equations.
Finding Common Denominators
When solving rational equations, finding a common denominator is crucial to combine terms effectively. First, identify the different denominators present in the equation. In our example, the denominators are \(x+4\), \(x-4\), and \(x^{2} - 16\). By noticing that \(x^{2} - 16\) factors to \((x + 4)(x - 4)\), we can use this factored form as the common denominator. Once you find the common denominator, you multiply each term in the equation by it. This process removes the denominators, allowing you to form a new equation that's easier to solve.
Simplifying Rational Expressions
Simplifying rational expressions involves combining like terms and reducing fractions. After clearing the denominators by multiplying each term by the common denominator, you will typically end up with a polynomial equation. For instance, when each term is multiplied by \((x + 4)(x - 4)\), we get: \[ 3(x-4) + 7(x+4) = 8\] Distributing and combining like terms inside the parentheses results in: \[ 3x - 12 + 7x + 28 = 8\] Combine terms to simplify further: \[ 10x + 16 = 8\] This process of simplifying helps in reducing the equation to a more straightforward form, making it easier to solve for the variable.
Checking Extraneous Solutions
After solving the equation, it's important to check for extraneous solutions. These are solutions that emerge from the steps of solving the equation but are not valid within the context of the original equation. To check for extraneous solutions, substitute your solution back into the original equation to ensure both sides are equal and that the solution does not make any denominator zero. For our equation, substituting \( x = - \frac{4}{5} \) back into the original expression confirms it does not cause any denominators to become zero, thereby verifying it as a valid solution. Remember, always check your solutions to ensure they fit within the constraints of the original problem.
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An extraneous solution to a rational equation is an algebraic solution that would cause any of the expressions in the original equation to be undefined. x + 1 3 = 5 6 . y + 2 3 = 1 5 . 3 + 1 5 = 1 x.
A rational function equation is of the form f(x) = P(x) / Q(x), where Q(x) ≠ 0. Every rational function has at least one vertical asymptote. Every rational function has at most one horizontal asymptote. Every rational function has at most one slant asymptote.
Using rational expressions and equations can help you answer questions about how to combine workers or machines to complete a job on schedule. A “work problem” is an example of a real life situation that can be modeled and solved using a rational equation.
The Rational Formula is expressed as Q = CiA where: Q =Peak rate of runoff in cubic feet per second C =Runoff coefficient, an empirical coefficient representing a relationship between rainfall and runoff.
A solution to an equation is a value of a variable that makes a true statement when substituted into the equation. The process of finding the solution to an equation is called solving the equation. To find the solution to an equation means to find the value of the variable that makes the equation true.
To find whether your solutions are extraneous or not, you need to plug each of them back in to your given equation and see if they work. It's a very annoying process sometimes, but if employed properly can save you much grief on tests or quizzes.
Step 1: Factor the numerator and the denominator. Step 2: List restricted values. Step 3: Cancel common factors. Step 4: Reduce to lowest terms and note any restricted values not implied by the expression.
We multiply the numerators and denominators of any two rational integers independently before simplifying the resultant fraction. To divide any two fractions, multiply the first fraction (dividend) by the reciprocal of the second fraction (which is the divisor).
A rational equation is an equation containing at least one fraction whose numerator and denominator are polynomials, \frac{P(x)}{Q(x)}. Q(x)P(x). These fractions may be on one or both sides of the equation.
Rational functions can be used in a variety of ways to model real-world situations. For example, we can use a rational function to model the speed of a car that's braking, or the amount of a drug in a person's system over time.
An asymptote is a line that a graph approaches but never touches. They are used in real life to help engineers and mathematicians design objects and predict how they will behave. For example, as a car moves faster and faster, the air resistance it experiences increases.
Two reasons that rational functions are important are that they arise naturally when we consider the average rate of change on an interval whose length varies and when we consider problems that relate the volume and surface area of three-dimensional containers when one of those two quantities is constrained.
Step 1: Factor the numerator and the denominator. Step 2: List restricted values. Step 3: Cancel common factors. Step 4: Reduce to lowest terms and note any restricted values not implied by the expression.
From the definition of a rational number that we talked about in the previous section, a rational number is of the form p/q, where 'p' and 'q' are integers and q≠0. Hence, the rational numbers formulas are: Q = {p /q: p, q belongs to Z; q ≠ 0}
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