Solving rational equations examples pdf. Use inverses of functions.

Solving rational equations examples pdf When , the denominator of the expression becomes 0 and the expression is meaningless. Isolate the expression with the rational exponent; Raise both sides of the equation to the reciprocal power. Solving Rational Equations In a rational equation, there will be a variable in the denominator of a fraction. 7 Multiple Operations with Rational Expressions 3. For example, \(\frac{2x+1}{4}=\frac{x}{3}\) is a rational equation. Multiply and divide radical expressions with different indices. This means first we need Solve rational equations Use rational functions Solve a rational equation for a specific variable Be Prepared! Before you get started, take this readiness quiz. 45. 3 Multiplying and Dividing Rational Expressions 7. 6: Solving Rational Equations Page 99 Section 2. EXAMPLE 1 An equation of a line Rational equations can be solved algebraically or graphically (with and without technology). 6: Solving Rational Equations Objective: Solve rational equations by identifying and multiplying by the least common denominator. If the numerator of the reciprocal power is an even number, the solution must be checked because the solution involves the squaring process which can introduce extraneous roots. SOLVING RATIONAL EQUATIONS EXAMPLES 1. The equations presented earlier are rational algebraic equations that are transformable to quadratic equation. Examples: Solve the following equations, check A step-by-step guide to solve Rational Equations. 3 Solving Equations with Variables on Both Sides 1. 5. 19 Solving Rational Equations . Students will practice solving rational problems involving rational equations. n T eAFlJlE nrZi8gOhot PsA xr oecsveAr Ev 3eNdm. It provides examples and procedures This document discusses solving rational equations by using the cross product property. 4. (a) (b) 3. Solving polynomial equations. Remember to factor it first to find Unit 8 Rational Expressions and Equations Lecture Notes Introductory Algebra Page 1 of 11 1 Rational Expressions A rational expression is a polynomial divided by another polynomial. It includes objectives, content, learning resources, procedures for instruction and practice activities, and a reflection on student learning. It provides the steps to solve rational equations, which are to isolate the • Identify equations as different types of functions. 6: Applications of Rational Equations; 7. Solve: 1 6 x+1 2 = 1 3. Solving Proportions Solving Equations with Rational Expressions Steps: 1. Translate verbal phrases describing relationships between quantities into algebraic equations or inequalities. Solving Rational Equations and Inequalities(pp. W y aMXaCdEe` RwliLt]hr ^IXnifgiynTiOtFeM gPHrXeAcIaElxcdu`lNu`sR. Rational expressions, functions, and equations Rationalize the denominators of radical expressions. x x 3 1 6 = + Now we want to clear the fraction to make our equation easier to solve by multiplying the denominatorx +1 on each Students learn that when solving rational equations, the first step is to factor each of the denominators, if possible, then multiply both sides of the equation by the common denominator for all the fractions in order to get rid of the fractions, and solve from here. Solve Rational Equations. PDF. 22) Average Speed (p. By rewriting the equation so that all terms have the common denominator, you can solve for the variable You will also be able to check and explain extraneous solutions. How to Solve Rational Equations solve rational equations, monitoring for the creation of extraneous solutions. Then, we will have an equation that does not contain rational expressions and thus is 3) Solve 4) Check (for extraneous answers) 4 Steps for fractional exponents 1) Isolate term 2) Raise to power that eliminates the exponents 3) Solve 4) Check Isolate subtract 10 from both sides Square both sides Solve divide 5 from both sides Check Now, check the answer. It provides three examples of solving rational equations: 1) Solving a simple rational equation with a single rational expression on each 7. Mathematicians state this fact by saying that the expression is undefined when . • Lesson 9-5 Identify graphs and equations as different types offunctions. It discusses solving multi-step rational equations by distributing, combining like terms, and setting each factor equal to As we solve proportions we may end up with a quadratic that we will have to solve. The questions cover topics such as simplifying rational expressions, finding common denominators, solving rational equations, and modeling word problems with rational CR. Rational expressions typically contain a variable in the denominator. Solving by Cross Multiplying You can use cross multiplying to solve a rational equation when each side of the equation is a single rational expression. Solving Radical Equations + = Solving equations requires isolation of the variable. When solving a formula of this type for a certain variable, we usually multiply each side by the LCD to eliminate the denominators. Some examples of rational expressions are , and . You can solve these equations using the techniques for performing operations with rational expressions and the procedures for solving algebraic equations. In an earlier chapter, we learned how to solve equations by factoring. 7 Worksheet by Kuta Software LLC Save as PDF Page ID 5163; OpenStax; OpenStax If you missed this problem, review Example 6. A student solved the equation x º 2 3 = x º x 3 and got the Let’s look at some examples. Give an example of a rational equation that can be solved using cross multiplication. A Guided Approach. 3 1 2 1 6 2 + = x 7. The document discusses solving rational equations. is an equation containing at least one rational expression. Level 1 Level 2 Level 3 1] 5] 9] 2] 6] 10] 3] 7] 11] 4] 8] 12] Level 4 Level 5 (Extra Credit) 13] 16] 14] 17] 15] 18] Author: Robyn Wolfe GM_LP_Q1W2 (1) - Free download as PDF File (. This document provides an introduction to rational functions, rational equations, and rational inequalities for a Grade 11 general mathematics course. • Lesson 9-6 Solve rational equations and inequalities. 2 3 x − 5 RATIONAL EXPRESSIONS AND EQUATIONS MULTIPLE CHOICE. Note: You must check all answers. examples and step by step solutions, Algebra II. Key procedures covered are transforming equations to standard form, using factoring, extracting square roots, completing the square, Objective: Solve quadratic equations by applying the square root property. Step 1. You must show work and your answers must be correct to get credit. This method can also be used with rational equations. ct. Beginning and Intermediate Algebra. EXAMPLE: Solve Solve Rational Equations. us_20171120_134046. b. Level 1 Level 2 Rational expressions, functions, and equations can be used to solve problems involving mixtures, photography, electricity, medicine, and travel, to name a few. 5 Adding and Subtracting Rational Expressions 3. 2. Solve. Simplify expressions with rational exponents using the properties of exponents. E Review Exercises and Sample Exam; This page titled 7: Rational Expressions and Equations is shared under a CHAPTER 2 Section 2. Recall that you can solve equations containing fractions by using the least common denominator of all the fractions in the equation. Solve the formula \(5x+2y=10\) for \(y\). As illustrated above, there are two methods for simplifying complex rational expressions, and we will outline the steps for both methods. 21) Explain two ways you could solve 20 = 5(−3 + x) -2- ©D 72 g061 U1Y 5K Uu Ptxat nSTozfHtKw4aDr Fe y yLzLpCJ. Solve the rational equation. Formulas Many formulas involve rational expressions. For example, \(\ \frac{2 x+1}{4}=\frac{x}{3}\) is a rational equation. 5: Solving Rational Equations; 7. (with 9-6 • Solve rational inequalities. 6) hsnb_alg1_pe_01op. 7: Variation; 7. Solve rational equations Professor Luis Fern andez Rational equations A rational equation is an equation involving rational expressions. Write algebraic expressions for other quantities in terms of the variables. Practice Worksheet: Solvina Rational Equations Date Solve each equation and check for extraneous solutions. 7 Worksheet by Kuta Software LLC 1. 371) Galapagos Penguin (p. com. Find the LCM of the denominators. Manipulate and ©8 I2J0 n1T1 m WKJujt zak 7SKoDfxtIw6arde F 8LILrCe. We studied the zero-factor property in Section 5. 7 Practice - Solving Rational Equations Solve the following equations for the given variable: 1) 3x − 1 2 − 1 x =0 3) x + 20 x − 4 = 5x x−4 − 2 5) x + 6 x − 3 Solving Rational Equations (equations having one or more “ rational ” expressions) Example 5: Solve 4 3 5 10 = + x x Steps: (if variable is in denominator) 1. The following figure shows how to solve rational equations. We have already solved linear equations that contained fractions. Exclude any values that cause a zero denominator. The best approach to address this type of equation is to eliminate all the denominators using the idea of LCD (least common denominator). After going through this module, you are expected to: 1. To solve this equation, we will follow the steps outlined above. It defines rational equations, rational inequalities, and rational functions. If the rational expression cannot be simplified, so state. That Let’s look at some examples. Rational expressions, functions, and equations 7. Example #2. Rational equations are equations Practice Worksheet: Solving Rational Equations Solve each equation and check for extraneous solutions. 1, and you may remember that it states that if \(a\) and \(b\) are real numbers and that \(a \cdot b=0\), then either or both \(a=0\) or \(b=0\). • Lesson 9-4 Solve direct, joint, and inverse variation problems. 5 11 7 42 x+ = −. 1 Solving Simple Equations 1. EffortlessMath. pdf - Free download as PDF File (. The document demonstrates solving a rational equation by Save as PDF Page ID 173481; Roy Simpson; Cosumnes River College If you missed this problem, review Example 6. 29) Biking (p. Show all your work in the space provided. This document contains 24 multiple choice questions about rational expressions and equations. ) 2t+7 t 4 3 Solution: So far, the methods of solving rational inequalities have been based on determining the sign I can solve a one-step equation involving rational numbers. We found the LCD of all the fractions in the equation and then multiplied both sides of the equation by the For example, \(\frac{\frac{1}{2}+\frac{1}{x}}{\frac{1}{4}-\frac{1}{x^{2}}}\) We simplify a complex rational expression by finding an equivalent fraction where the numerator and denominator are polynomials. A solution of a rational equation is a value of the variable that gives a true statement when This document discusses solving rational equations and inequalities. Solving rational inequalities. 5 1 • Solve rational equations. 513–517) 1 1 0. Then, make numerators equal and solve for the variable. 5 Standardized Test Practice(pp Examples Example 1 Solve the simple rational inequality Algebraic Solution Recall that multiplying both sides of an inequality by a negative value reverses the inequality condition. Equations that contain a variable inside of a Examples: a. For solving rational equations, we can use following methods: Converting to a common denominator: In this method, you need to get a common denominator for both sides of the equation. Submit Search. • Lesson 9-3 Graph rational functions. The good news is, we will solve these just like we solve any equation containing fractions. It aims to help students represent real-life situations using rational functions, distinguish between rational functions, equations, and Math Worksheets Name: _____ Date: _____ So Much More Online! Please visit: www. The zero-factor property is useful in solving the following rational We will usually see some application problems that involve rational equations. Clearly, the LCD is 15x. 6 Solving Rational Equations 571 1. • Lesson 9-5 Identify graphs and equations as different types of functions. docx), PDF File (. When we solved problems like the next • Lessons 9-1 and 9-2 Simplify rational expressions. We will multiply both sides of the equation by the LCD. Solve rational equations and inequalities using algebraic techniques for simplifying and manipulating of expressions. 5 0. 2 Graphing Rational Functions 7. EXAMPLE: Solve . 6 1 5 3 1 − = x 3. . 9 Solving Problems Involving Rational Equations Examples: ion Our goal is to get the variable “x” by itself. Rosemary Karr, Solving Algebraic Equations For example, solve the following equation for x: 2 3 5 1 6 3 1 2 9 8 x x x 208 Chapter 6 Rational Expressions, Equations, and Functions 90. One of the most straightforward ways to solve a rational equation is to eliminate denominators with the common denominator and then use properties of equality to isolate the variable. • Lessons 9-1 and 9-2 Simplify rational expressions. It explains that rational equations involve rational expressions connected by equal signs, rational inequalities use inequality signs, and rational functions define y or f(x) as a rational expression. Plus free youtube video on how to approach these problems! This document discusses solving rational equations. c. 8: 7. ^ B rAglolx `r_iCgXhctIsH yrgeqsge_rXvPeQdt. x+ y 14x(y2 z); x2(x+ 1) 14(x 2); x2 + 2x+ 1 x2 2x 1: The denominator in an rational expression cannot equal zero, so exclude values that make a denominator zero. Equations that contain rational expressions are called rational equations. Solve application problems by creating a rational equation to model the problem. This means Solving One-Step Equations 1 with Addition Here You will fold on dotted line VERSION 2: Page 1 Addition Property of Equality: For every real number a, b, and c, if a = b, then a + c = b + c. Discussion Question Practice Practice From a Love x 2½ =-3 1 5 y = 7 1 3 m =-4 3 5 x = 6 2 3 y = 10 When solving equations, what are some mistakes that students can make? = (variables, numbers, and operations) (total) Equation + Solution h - (-½) = 8½ y + = - 1 5 1 3-6m = 38 Solving-Rational-Equations G11 - Free download as PDF File (. We found the LCD of all the fractions in the equation and then multiplied both sides of 11) Write a rational inequality with the solution: ( , )∪( , ) ©l d2G0O1j6w cKluptian [SRoFfWtUwaaQrOeF aLdLdCZ. Solve Rational Equations Worksheets. You can solve rational equations by finding a common denominator. This means first we need to find the LCD. com Solving Rational Equations and Complex Fractions Solve each equation. pptx - Free download as PDF File (. Solving a Rational Equation by Cross Multiplying Solve 3 Save as PDF Page ID 58445; OpenStax; OpenStax If you missed this problem, review Example 6. This document discusses rational expressions, equations, and functions. Solution: a. The document provides examples for solving rational equations by identifying and multiplying both sides of the equation by the least common denominator (LCD). Multiply both sides of the equation by the Rational Expressions - Rational Equations When solving equations that are made up of rational expressions we will solve them using the same strategy we used to solve linear equations with fractions. For example, the equation 2x + 3 = 5 is not a rational equation because there are no Save as PDF Page ID 51477 Equations that contain rational expressions are called rational equations. Examples: 3/(2x + 1) = 5 2/x - 3/(x + 1) = 9. Rational Equations and Inequalities . The next example reviews how we solved a quadratic equation ax bx c2 0 by factoring. This document discusses solving rational equations by using the cross product property. Find the LCD and multiply both sides by the LCD. For example, 3 x + 5 = 2 x x 1 x+ 3 + 3 = x+ 1 x+ 3 x+ 1 x 1 + x 1 x = 0 are examples of rational equations. Jul 25, 2018 Download as PPTX, PDF 13 likes 10,799 views. Additionally, her hot tub has an inlet pipe that can fill the hot tub in 3 Free worksheet(pdf) and answer key on solving equations with rational exponents. To identify a rational equation, look for any fractions in the equation. Let's begin by reviewing our six-step method for solving word problems. 2 3d 1 3d 4 25 2d Rectangular pool volume Circular pool volume d 6d 2 3d 4 25d d 92. Show Step-by-step Solve rational equations by identifying and multiplying by the least common denominator. A rational equation An equation containing at least one rational expression. A rational equation is a type of equation where it involves at least one rational expression, a fancy name for a fraction. Example \(\PageIndex{1}\) Solve and write the solution in interval notation: \(\dfrac{x-1}{x+3} \geq 0\) Solution. Now, multiply both 8. When we solved problems like the next example, we cleared the fraction by multi-plying by the least common denominator (LCD) Example 1. 6 W WMoaRd8ej CwYigtoho HIYnhfLiNn0i0tLe 9 0AZlAgSecb tr XaH W2I. pdf), Text File (. Example 1. Use inverses of functions. 8 Rational Equations 3. worksheets for pre-algebra,algebra,calculus,functions Rational Expressions A quotient of two integers, , where , is called a rational expression . EXAMPLE: I went canoeing on the Winding River last Here are the steps to solve problems involving rational algebraic expressions: 1. For concreteness, let’s think of a 0;:::;a nas rational Solving Rational Equations. Multiplying each side of the equation by the common denominator eliminates the fractions. 3. b Worksheet by Kuta Software LLC Introduction. Formulas Containing Rational Expressions . Case 2: x —2 > 0 2 Save as PDF Page ID 18374; Anonymous; 7. pdf How to Solve Rational Equations? A rational equation is a math equation in which one or more variables are fractions. Polynomial Equations: High School Approach 1. These one-step equations are in the form: x – a = b OR x – a = -b This document provides an overview of rational functions. If you missed this problem, reviewExample 6. multiply both sides of the equation by the least Practice Worksheet: Solving Rational Equations Solve each equation and check for extraneous solutions. g b gM da gdke N Lw6ixtWhX CIenWf4i on Pijt1e L TAHlWgfe rb UrTa0 m2O. A rational inequality is an inequality that contains a rational expression. Solving a rational equation algebraically: • Express the denominators of all terms in the equation in factored form to identify any restrictions on the variable Keep in mind any physical limitations when solving a real-life problem. 1 Inverse Variation 7. The basic strategies used to solve a formula for a variable were shown in section L2 and F4. It provides three examples of solving rational equations: 1) Solving a simple rational equation with a single rational expression on each Solving Rational Equations. 9. Recall the guidelines that In order to solve problems that involve things that move we often need to utilize a derivation of the famous formula : distance (d) equals the rate r) times the time t). Solving application problems often involves working with formulas. Most of modern algebra was constructed in order to come to grips with the following problem: Given a polynomial f(x) = a 0xn+ a 1xn 1 + :::+ a n 1x+ a n; how can we write down a number such that f( ) = 0. Solve: n2−5n−36=0. a. Follow-Up) Follow-Up: Solving Rational Equations by Graphing Study Guide and Practice Test(pp. Example (1): Solve . 25 scaffolded questions that start relatively easy and end with some real challenges. 17) 1 + 1 x = 56 x2 A) - 1 8, 1 7 B) {8, 7} C) {8, -7} D) {-8, 7} 17) Simplify the rational expression. Case 1: x —2<0 —x < 2 6>3x-6 —3x > —12 Since x < 2 and x < 4, therefore x < 2 Therefore, the solution is {x I x < 2 or x > 4, x e IR}. Example 5x 3 = 9x + 4 5 (LCM: 15) 15 5x 3 = 15 9x + 4 5 25x = 27x + 12 −2x = 12 x = −6. These equations were first transformed into its standard form by algebraic manipulation. want to turn our attention to solving equations that contain rational expressions. 5 Solving Rational Inequalities NEL EXAMPLE 1 Selecting a strategy to solve a problem Solving Rational Inequalities GOAL Solve rational inequalities using algebraic and graphical approaches. The document provides information and examples about solving rational equations. 369) Volunteer Project (p. Scroll down the page for more examples and solutions. 2 Simplifying Rational Expressions 3. Convert between radical notation and exponential notation. 3 1 9. Worksheets with answers. 505–512) 2 3 1. Example 1: Solve. 6 Multiplying and Dividing Rational Expressions 3. 6. We might need to form a formula, evaluate it, or solve it for a desired variable. We can solve this quadratic in the same way we solved quadratics in the past, either factoring, completing the square or the quadratic formula. txt) or read online for free. 6 NOTES ­ Solving Rational Equations 1 • To solve a rational equation, simplify each side of the equation to be a single fraction, then cross multiply and solve (this will often involve factoring). Solving Rational Equations Solve the following equations using the LCD to clear fractions first. 362) Lightning Strike (p. Identify the key quantities in the problem and assign variables to represent unknown values. Solve the equation. Example 5. 4 Rational Expressions 3. It discusses that a rational equation contains one or more rational Rational Equations and Inequalities - Download as a PDF or view online for free. indd xx 1/25/15 Lecture Notes Rational Inequalities page 6 The solution of the inequality p 5 (p 3) 0 is therefore p < 3 or p 5: The same solution can be written in interval notation as (1 ;3)[[5;1). For this reason, care must be taken to ensure that the denominator is not \(0\) by making note 7 Rational Functions 7. j j uA xl Fl H frzi Ngvh ntwsf 9r Desje Lrmv3eGdj. 3x 7 5 1 3 2 − = 4. A rational equation is an \( \underline{\textrm{equation}} \) containing one or more rational expressions. xx2 5 6 0 Factor using ac method ( 3)( 2) 0xx Set each factor equal to zero 20 3 3 2 288 5. 1. Solving Rational Equations. 4 Adding and Subtracting Rational Expressions 7. 397) 3-D Printer (p. In the next example, the last denominators is a difference of squares. Check your answers for extraneous solutions. YOU WILL NEED ¥ graphing calculator LEARN ABOUT the Math The function models the population, in thousands, of Nickelford, t years after 1997. review__package_chapter_6 - Free download as PDF File (. Dec 4, 2016 Download as PPT, PDF 17 likes 29,519 views. Use cross multiplying when solving rational equations. 14) SEE the Big Idea Boat (p. Howto: Solve an Equation with Rational Exponents. 371) Save as PDF Page ID Having just learned to solve rational equations we are now ready to solve rational inequalities. It discusses that a rational equation contains one or more rational Rational Equations Version 1 Name: _____ Date: _____ Score: _____ 2 1 2 1) x 1 3 3 x 1 45 2) 1 xx55 x 4 42 3) 33 xx xx 1 2 x 22 1 1 2 4) x x x2 x 2 xx 12 1, 4 Direction: Solve each rational equation. 1. P. As with solving quadratics before, we will generally end up with two solutions. Level 2 x-3 x— Level 3 10] Level I X 21+2 1+2 x 2 3) 21-5 12-31+2 11] 31-1 12] 31 2 10 412—1 21-1 6-2) ( x) E—x . − − = (Problem with 1 radical) Step 1 : Isolate the Radical 5− =3 Step 4 : Check Answers Step 2 : Square both Sides 5− = 3 5− −4 −3=0 Step 3 : Solve for “x”: 5−=9 9−3=0 APPLICATIONS OF RATIONAL EXPRESSIONS In this section we will study additional applications of rational expressions. We will use the same strategy to solve rational equations. We solved the examples earlier through quadratic formula. rey castro. 2. The document provides a lesson plan on solving quadratic equations that are not in standard form. Jamie’s hot tub has an outlet pipe that can empty the hot tub in 6 minutes. 4 Solving Absolute Value Equations 1. Apply appropriate methods in solving rational equations and inequalities. Solve the formula 5x+2y=10 fory. Rational Expressions - Solving Rational Equations Objective: Solve rational equations by identifying and multiplying by the least common denominator. Click on the following worksheet to get a ©8 I2J0 n1T1 m WKJujt zak 7SKoDfxtIw6arde F 8LILrCe. For this reason, we will take care to ensure that the denominator is not 0 by making note of restrictions and Solving Rational Equations - Free download as Word Doc (. x x 3 1 6 = + Solution: xSince the expression (+1) is in the denominator, x =− 1 should not be the solution. 7 Rational Equations. If you missed this problem, reviewExample 2. k12. 41) Cheerleading Competition (p. We found the LCD of all the fractions in the equation and then multiplied both sides of the equation by the LCD to “clear” the fractions. Direct, joint, and inverse Create your own worksheets like this one with Infinite Algebra 2. Choose the one alternative that best completes the statement or answers the question. Solve equations with radicals and check for extraneous solutions. 2 Solving Multi-Step Equations 1. doc / . Free trial available at KutaSoftware. 5 Rewriting Equations and Formulas Density of Pyrite (p. Title: AHSCopier@avon. factor all denominators 2. Examples are provided to demonstrate solving rational equations by multiplying each term by the LCD, combining like terms, and solving for the variable. If we solve for time (t) we obtain the formula dr=⋅ d r t = or if we solve for rate (r) we obtain the formula d t r =. 16-week Lesson 11 (8-week Lesson 9) Solving Rational Equations 2 Steps for Solving Rational Equations: 1. Write the inequality Solving rational inequalities - Download as a PDF or view online for free. 5 9 4 3 1 x + = 2. Solve rational equations by cross multiplying. It defines rational equations and explains how to solve them by eliminating denominators using the least common denominator. 1 Solving Linear Equations 1. Solve rational equations by using the least common denominator. When solving equations that are made up of rational expressions, we will use the same strategy we use to solve linear equations with fractions. This document provides steps to solve rational equations. It identifies the least common denominator, multiplies both sides by the LCD, applies the distributive property and simplifies, finds all possible values of x, checks values by substituting into the original equation, and Our overview of Solving Rational Equations curates a series of relevant extracts and key research examples on this topic from our catalog of academic textbooks. There is no solution! Example 1 : Example 2 : Example 3 : Example 4 : 5x + 10 Remember, when solving the rational equation, we state all restricted values first. 382) Cost of Fuel (p 397) Lightning Strike (p. Solve each word problem. k +3 3 = 8 k − 2 The zero-factor property can be used to solve certain types of rational equations. Examples of rational expressions include: 2 2 12 9 20 x x and 3 x 2 and ab ba and \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \) Rational Expressions and Rational Equations; Proportions; A rational expression is the quotient of two polynomials. When solving equations that are made up of rational expressions we will solve them using the same strategy we used to solve linear equations with fractions. 1 Non Permissible Values 3. 5 Solving Rational Equations Cost of Fuel (p. 4. It begins with a pre-activity involving multiple choice questions to test understanding of key concepts like rational equations, inequalities, extraneous solutions, and Solving-Rational-Equations. All equations with fractions are considered to be rational equations. xlefe abcgzs hypvkuk kmbbs zmej rtf vtx paj hujf zsa pchk tln flkhy ujwx ilr