Overview of Section 5. Introduction In this chapter, students have worked with nth roots, writing them in radical form and with rational exponents. The learned that the properties of exponents can be applied to rational exponents. The have also examined the graphs of radical functions. In this lesson, students will solve radical equations, beginning with square root equations. Given past work with solving equations in one variable, students should be ver comfortable with performing the same operations to both sides of an equation. All the steps taken in the past to solve equations were reversible, meaning ou start with one (true) equation and end up with an equivalent (true) equation. Squaring both sides of an equation is a tpical example of an irreversible step. You can begin with a statement that is not true, such as, and square both sides to end up with a true equation, ( ) =. Working with radical equations, x = has no solution, but after squaring both sides of the equation ou obtain the statement x = 9. In this lesson, it is necessar to check apparent solutions for each problem because extraneous solutions ma be introduced b raising each side of the equation to an exponent. Dnamic Teaching Tools Dnamic Assessment & Progress Monitoring Tool Lesson Planning Tool Interactive Whiteboard Lesson Librar Dnamic Classroom with Dnamic Investigations Formative Assessment Tips Think-Alouds: This technique is used when ou want to hear how well partners comprehend a process involved with solving a problem. It is important to model the process first so that students have a sense of what is expected in Think-Alouds. Think-Alouds give students the opportunit to hear the metacognitive processes used b someone who is a proficient problem solver. Make Sense of Problems and Persevere in Solving Them and Attend to Precision: Hearing someone else describe a process using mathematical language will improve all students problem-solving abilities; the can now appl the process with their partners to the problem being solved. Use this technique with a multistep problem. Model using a starter sentence such as: The problem is asking, I can use the strateg of, The steps I will use in solving this problem are, This problem is similar to, and I can check m answer b. Use this technique for a variet of problem tpes. Listen for comprehension of skills, concepts, and procedures as well as precision of language. Another Wa It is possible to solve man of the radical equations graphicall. This can be done with a graphing calculator b treating each side of the equation as a function. Graph each function and look for point(s) of intersection. Pacing Suggestion The formal lesson is quite long. You might have students do onl a few of the matching problems from Exploration and then spend some time discussing numerical and analtical approaches to equation solving in Exploration. Section 5. T-0
What Your Students Will Learn Solve equations containing radicals and rational exponents and check for extraneous solutions. Solve radical inequalities b first solving for the variable and then considering the domain of the radicand. Exploration Motivate Students are likel well aware of different levels of hurricanes. A categor hurricane has wind speeds of 7 to 95 miles per hour and air pressure of 980 millibars or more. A millibar (mb) is a measure of the pressure (or weight) of the air that is usuall taken as close to the core of the hurricane as possible. As a general rule, the lower the pressure, the higher the winds. Explain to students that in this lesson the will solve an equation involving wind velocit and air pressure. If possible, share some statistics about a hurricane that our students might be familiar with, such as Hurricane Sand or Hurricane Katrina. Discuss Students are not expected to have a sense of what the graphs of these equations look like. The should, however, have a sense of how to explore the equations b looking at ke values of x. Model this b discussing the first equation in the following manner: Sa, When equation (a) is evaluated at x =, the result is = 0, which is never true. There is onl one graph that looks like it does not include x = in the domain, and that is graph (C). Evaluating the same equation at x = 0 and x = confirms that this equation matches graph (C), because the function is undefined at x = 0 and when x =, = 0. The x-intercept is, so this is the solution for equation (a). Exploration Model what it sounds like to test ke values of x in the equation. Think-Alouds: Select another equation and sa, I should evaluate this equation when x =. Ask partner B to think aloud for partner A to hear the reasoning process. When students have finished the problem, select a new equation and reverse roles. Not all of the equations are equal in difficult, so let students select the equations that trigger sense making for them. The might also look at the graphs and think, Graphs (A) and (B) have range values onl in Quadrants I and II. Which equations have onl positive range values? Exploration Think-Pair-Share: Depending on how much time is spent on Exploration, ou might choose for students to do a Think-Pair-Share for this exploration. It is the discussion that is important versus actuall solving a particular radical equation. Communicate Your Answer Listen for an understanding of the various approaches. The understanding would indeed appl to other tpes of equations, not just radical equations. Connecting to Next Step In the formal lesson, students will use an analtic approach to solve radical equations. Having a sense of how to solve the equations graphicall will be helpful to students. T- Chapter 5
5. Solving Radical Equations and Inequalities Essential Question How can ou solve a radical equation? Solving Radical Equations Work with a partner. Match each radical equation with the graph of its related radical function. Explain our reasoning. Then use the graph to solve the equation, if possible. Check our solutions. a. x = 0 b. x + x + = 0 c. 9 x = 0 d. x + x = 0 e. x + x = 0 f. x + = 0 A. C. E. B. D. F. Dnamic Teaching Tools Dnamic Assessment & Progress Monitoring Tool Lesson Planning Tool Interactive Whiteboard Lesson Librar Dnamic Classroom with Dnamic Investigations. a. C; The domain of the equation is x ; x = b. E; The domain of the equation is x ; x = c. B; The domain of the equation is x ; x = ± d. D; The domain of the equation is x ; x = e. F; The domain of the equation is x ; x = f. A; The domain of the equation is all real numbers; no real solutions. a. Sample answer: x 0 9 x 0 5 8 8 5 0 LOOKING FOR STRUCTURE To be proficient in math, ou need to look closel to discern a pattern or structure. Solving Radical Equations Work with a partner. Look back at the radical equations in Exploration. Suppose that ou did not know how to solve the equations using a graphical approach. a. Show how ou could use a numerical approach to solve one of the equations. For instance, ou might use a spreadsheet to create a table of values. b. Show how ou could use an analtical approach to solve one of the equations. For instance, look at the similarities between the equations in Exploration. What first step ma be necessar so ou could square each side to eliminate the radical(s)? How would ou proceed to find the solution? Communicate Your Answer. How can ou solve a radical equation?. Would ou prefer to use a graphical, numerical, or analtical approach to solve the given equation? Explain our reasoning. Then solve the equation. x + x = Section 5. Solving Radical Equations and Inequalities The solutions are x = and x =. b. Sample answer: 9 x = 0 9 x = 0 x = ± Isolate the radical; square both sides.. Isolate the radical, and then square both sides.. Sample answer: graphical; The graph shows the solution; x = Section 5.
Differentiated Instruction Visual Have students create a flow chart summarizing the steps involved with solving radical equations. Direct them to use the Core Concept as a guide. Extra Example a. Solve x + = 8. x = b. Solve x 5 =. x = 5 5. Lesson What You Will Learn Core Vocabular radical equation, p. extraneous solutions, p. Previous rational exponents radical expressions solving quadratic equations Solve equations containing radicals and rational exponents. Solve radical inequalities. Solving Equations Equations with radicals that have variables in their radicands are called radical equations. An example of a radical equation is x + =. Core Concept Solving Radical Equations To solve a radical equation, follow these steps: Step Isolate the radical on one side of the equation, if necessar. Step Raise each side of the equation to the same exponent to eliminate the radical and obtain a linear, quadratic, or other polnomial equation. Step Solve the resulting equation using techniques ou learned in previous chapters. Check our solution. MONITORING PROGRESS. x = 7. x =. x = Check + =? =? = Solving Radical Equations Solve (a) x + = and (b) x 9 =. a. x + = Write the original equation. x + = Divide each side b. ( x + ) = Square each side to eliminate the radical. x + = Simplif. x = Subtract from each side. The solution is x =. b. x 9 = x 9 = Write the original equation. Add to each side. Check (8) 9 =? 7 =? = ( x 9 ) = Cube each side to eliminate the radical. x 9 = 7 Simplif. x = Add 9 to each side. x = 8 Divide each side b. The solution is x = 8. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Solve the equation. Check our solution.. x 9 =. x + 5 =. x = Chapter 5 Rational Exponents and Radical Functions Teacher Actions Big Idea: Ask students how the would solve x 5 = 7. Then ask how the would solve x 5 = 7. The steps are the same. Isolate the radical term, x =. Square both sides to solve. Think-Alouds: Model solving Example (a). Pose Example (b) and sa, To solve this problem I need to. Ask partner A to think aloud for partner B to hear the problem-solving process. Popsicle Sticks: When students have finished the example, use Popsicle Sticks to solicit responses. Chapter 5
ATTEND TO PRECISION To understand how extraneous solutions can be introduced, consider the equation x =. This equation has no real solution; however, ou obtain x = 9 after squaring each side. Solving a Real-Life Problem In a hurricane, the mean sustained wind velocit v (in meters per second) can be modeled b v( p) =. 0 p, where p is the air pressure (in millibars) at the center of the hurricane. Estimate the air pressure at the center of the hurricane when the mean sustained wind velocit is 5.5 meters per second. v( p) =. 0 p Write the original function. 5.5 =. 0 p Substitute 5.5 for v( p). 8.5 0 p Divide each side b.. 8.5 ( 0 p ) Square each side. 7.8 0 p Simplif. 98. p Subtract 0 from each side. 98. p Divide each side b. The air pressure at the center of the hurricane is about 98 millibars. Monitoring Progress Help in English and Spanish at BigIdeasMath.com. WHAT IF? Estimate the air pressure at the center of the hurricane when the mean sustained wind velocit is 8. meters per second. Raising each side of an equation to the same exponent ma introduce solutions that are not solutions of the original equation. These solutions are called extraneous solutions. When ou use this procedure, ou should alwas check each apparent solution in the original equation. Solve x + = 7x + 5. Solving an Equation with an Extraneous Solution x + = 7x + 5 Write the original equation. (x + ) = ( 7x + 5 ) Square each side. x + x + = 7x + 5 x 5x = 0 (x 7)(x + ) = 0 Factor. Expand left side and simplif right side. Write in standard form. x 7 = 0 or x + = 0 Zero-Product Propert x = 7 or x = Solve for x. English Language Learners Vocabular Write the word extraneous on the board and circle the prefix extra-. Point out that this prefix means outside of or in addition to. Explain that an extraneous solution is a solution that is outside of, or in addition to, the actual solution set. Extra Example In a hurricane, the mean sustained wind velocit v (in meters per second) can be modeled b v(p) =. 0 p, where p is the air pressure (in millibars) at the center of the hurricane. Estimate the air pressure at the center of the hurricane when the sustained wind velocit is. meters per second. The air pressure at the center of the hurricane is about 959. millibars. Extra Example Solve x + 7 = x. The apparent solution x = is extraneous. So, the onl solution is x = 9. MONITORING PROGRESS ANSWER. about 95 millibars Check 7 + =? 7(7) + 5 + =? 7( ) + 5 8 =? =? 8 = 8 The apparent solution x = is extraneous. So, the onl solution is x = 7. Section 5. Solving Radical Equations and Inequalities Teacher Actions Think-Alouds: Pose Example and sa, This example is similar to Example (a), so I need to Ask partner B to think aloud for partner A to hear the similarit of the problem-solving process. Popsicle Sticks: When students have finished the example, use Popsicle Sticks to solicit responses. If students have been checking solutions for each problem, it will seem natural to do so for Example. In Example, wh do ou think we got an answer of that is not a solution? Students ma see that squaring resulted in an x -term. Discuss extraneous solutions. Section 5.
Extra Example Solve x + + = 7 x. The apparent solution x = is extraneous. So, the onl solution is x =. Extra Example 5 Solve (x) / = 5. The solution is x = 7. MONITORING PROGRESS 5. x = and x = 0. x = 7. x = ANOTHER METHOD You can also graph each side of the equation and find the x-value where the graphs intersect. Solve x + + = x. Solving an Equation with Two Radicals x + + = x Write the original equation. ( x + + ) = ( x ) Square each side. x + + x + + = x Expand left side and simplif right side. x + = x Isolate radical expression. x + = x Divide each side b. ( x + ) = ( x) Square each side. x + = x Simplif. 0 = x x Write in standard form. 0 = (x )(x + ) Factor. x = 0 or x + = 0 Zero-Product Propert x = or x = Solve for x. Check + + =? + + =? ( ) Intersection X=- Y= + =? + =? = The apparent solution x = is extraneous. So, the onl solution is x =. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Solve the equation. Check our solution(s). 5. 0x + 9 = x +. x + 5 = x + 7 7. x + = x When an equation contains a power with a rational exponent, ou can solve the equation using a procedure similar to the one for solving radical equations. In this case, ou first isolate the power and then raise each side of the equation to the reciprocal of the rational exponent. Solve (x) / + = 0. Solving an Equation with a Rational Exponent (x) / + = 0 Write the original equation. (x) / = 8 Subtract from each side. [(x) / ] / = 8 / Raise each side to the four-thirds. x = Simplif. x = 8 Divide each side b. The solution is x = 8. Check ( 8) / + =? 0 / + =? 0 0 = 0 Chapter 5 Rational Exponents and Radical Functions Teacher Actions Pose Example and ask, Is there a wa to combine the radical terms? no Turn and Talk: Work through the example. What would happen if ou got both radical terms on the same side of the equation before squaring in the first step? Squaring a binomial with two radical terms produces two terms that have unlike radical terms. Look For and Make Use of Structure: Before Example 5, make the connection between x + = and (x + ) / =. We are using the Power of a Power Propert. Chapter 5
Check ( + 0) / =? / =? = ( 5 + 0) / =? 5 5 / =? 5 5 5 Solve (x + 0) / = x. Solving an Equation with a Rational Exponent (x + 0) / = x Write the original equation. [(x + 0) / ] = x x + 0 = x Square each side. Simplif. 0 = x x 0 Write in standard form. 0 = (x )(x + 5) Factor. x = 0 or x + 5 = 0 Zero-Product Propert x = or x = 5 Solve for x. The apparent solution x = 5 is extraneous. So, the onl solution is x =. Monitoring Progress Solve the equation. Check our solution(s). Help in English and Spanish at BigIdeasMath.com 8. (x) / = 9. (x + ) / = x 0. (x + ) / = 8 Solving Radical Inequalities To solve a simple radical inequalit of the form n u < d, where u is an algebraic expression and d is a nonnegative number, raise each side to the exponent n. This procedure also works for >,, and. Be sure to consider the possible values of the radicand. Solve x. Solving a Radical Inequalit Extra Example Solve (x + ) / = x. The apparent solution x = is extraneous. So, the onl solution is x =. Extra Example 7 Solve x 8. The solution is 0 x. MONITORING PROGRESS 8. x = 9 9. x = 0. x =. a. x 9 b. x < 7 Step Solve for x. x Write the original inequalit. Check 0 = x Divide each side b. x Square each side. x 7 Add to each side. Intersection X=7 Y= 8 = x Step Consider the radicand. x 0 x So, the solution is x 7. The radicand cannot be negative. Add to each side. Monitoring Progress Help in English and Spanish at BigIdeasMath.com. Solve (a) x and (b) x + < 8. Section 5. Solving Radical Equations and Inequalities 5 Teacher Actions Think-Alouds: Pose Example and sa, This example is similar to Example 5, so I need to Ask partner B to think aloud for partner A to hear the similarit of the problem-solving process. Popsicle Sticks: When students have finished the example, use Popsicle Sticks to solicit responses. Step in Example 7 ma seem unnecessar to students. The graphical solution should help. The domain of the radical equation is x. Closure Writing Prompt: It is necessar to check apparent solutions when solving radical equations because Sample answer: raising each side of an equation to the same exponent ma introduce solutions that are not solutions to the original equation. Section 5. 5
Assignment Guide and Homework Check ASSIGNMENT Basic:,, odd, 5, 58, 70 Average:,, 50 even, 5 58 even, 70 Advanced:,, even,, 58 even, 59 70 HOMEWORK CHECK Basic: 5,, 7, 7, Average:,,,, Advanced: 0,,,,. no; The radicand does not contain a variable.. First, subtract 0 from both sides of the inequalit. Then square each side. Eliminate an solutions that would make the radicand negative.. x = 7. x = 8 5. x =. x = 7 7. x = 8. x =. 9. x = 000 0. x = 9. x = 0. no real solution. about.7 r. about.5 m 5. x =. x = 5 7. x = 8. x = 9. x = 0 and x = 0. x =. x =. x = 7.5. x =. x = 5 5. x =. x = 0.5 7. x =±8 8. x = 9. no real solution 0. x = 8. x = 5. Exercises Dnamic Solutions available at BigIdeasMath.com In Exercises, solve the equation. Check our solution. (See Example.). 5x + =. x + 0 = 8 5. Vocabular and Core Concept Check. VOCABULARY Is the equation x = a radical equation? Explain our reasoning.. WRITING Explain the steps ou should use to solve x + 0 < 5. Monitoring Progress and Modeling with Mathematics x =. 7. x + = 8. 8 0x 5 = 7 x 0 = 7 9. x + 0 = 8 0. x 5 = 0 5. x + 7 = 5. x = 5. MODELING WITH MATHEMATICS Biologists have discovered that the shoulder height h (in centimeters) of a male Asian elephant can be modeled b h =.5 t + 75.8, where t is the age (in ears) of the elephant. Determine the age of an elephant with a shoulder height of 50 centimeters. (See Example.). MODELING WITH MATHEMATICS In an amusement park ride, a rider suspended b cables swings back and forth from a tower. The maximum speed v (in meters per second) of the rider can be approximated b v = gh, where h is the height (in meters) at the top of each swing and g is the acceleration due to gravit (g 9.8 m/sec ). Determine the height at the top of the swing of a rider whose maximum speed is 5 meters per second. h Chapter 5 Rational Exponents and Radical Functions. x =. x = 5. x = and x = 5. Onl one side of the equation was cubed; x 8 = ( x 8 ) = x 8 = x = In Exercises 5, solve the equation. Check our solution(s). (See Examples and.) 5. x = x. x 0 = 9x 7. x = x 0 8. x + 0 = x + 9. 8x = x 0. 8x = x. x + = x + 0. x x + = 0.. x 5 8x + = 0 x + 5 = x + 5. x 8 + = x + 5. x + = x In Exercises 7, solve the equation. Check our solution(s). (See Examples 5 and.) 7. x / = 8 8. x / = 9. x / + = 0 0. x / = 0. (x + ) / = x. (5 x) / x = 0. (x + ) / = x +. (5x ) / = x ERROR ANALYSIS In Exercises 5 and, describe and correct the error in solving the equation. 5. x 8 =. ( x 8 ) = x 8 = x = x = 8x / = 000 8(x / ) / = 000 / 8x = 00 x = 5. When raising each side to an exponent, the 8 was not included; 8x / = 000 (8x / ) / = 000 / x = 000 x = 5 Chapter 5
In Exercises 7, solve the inequalit. (See Example 7.) 7. x 5 8. x 5 9. x > 0 0. 7 x + < 9. x + 8. x + 7. x + <. 0.5 x 5. MODELING WITH MATHEMATICS The length (in inches) of a standard nail can be modeled b = 5d /, where d is the diameter (in inches) of the nail. What is the diameter of a standard nail that is inches long?. DRAWING CONCLUSIONS Hang time is the time ou are suspended in the air during a jump. Your hang time t (in seconds) is given b the function t = 0.5 h, where h is the height (in feet) of the jump. Suppose a kangaroo and a snowboarder jump with the hang times shown. t = 0.8 t =. a. Find the heights that the snowboarder and the kangaroo jump. b. Double the hang times of the snowboarder and the kangaroo and calculate the corresponding heights of each jump. c. When the hang time doubles, does the height of the jump double? Explain. USING TOOLS In Exercises 7 5, solve the nonlinear sstem. Justif our answer with a graph. 7. = x 8. = x + 7 = x = x + 5 9. x + = 50. x + = 5 = x = x + 5 5. x + = 5. x + = = x = x + 5. PROBLEM SOLVING The speed s (in miles per hour) of a car can be given b s = 0 fd, where f is the coefficient of friction and d is the stopping distance (in feet). The table shows the coefficient of friction for different surfaces. Surface Coefficient of friction, f dr asphalt 0.75 wet asphalt 0.0 snow 0.0 ice 0.5 a. Compare the stopping distances of a car traveling 5 miles per hour on the surfaces given in the table. b. You are driving 5 miles per hour on an ic road when a deer jumps in front of our car. How far awa must ou begin to brake to avoid hitting the deer? Justif our answer. 5. MODELING WITH MATHEMATICS The Beaufort wind scale was devised to measure wind speed. The Beaufort numbers B, which range from 0 to, can be modeled b B =.9 s +.5.55, where s is the wind speed (in miles per hour). Beaufort number Force of wind 0 calm gentle breeze strong breeze 9 strong gale hurricane a. What is the wind speed for B = 0? B =? b. Write an inequalit that describes the range of wind speeds represented b the Beaufort model. 55. USING TOOLS Solve the equation x = x. Then solve the equation x = x. a. How does changing x to x change the solution(s) of the equation? b. Justif our answer in part (a) using graphs. 5. MAKING AN ARGUMENT Your friend sas it is impossible for a radical equation to have two extraneous solutions. Is our friend correct? Explain our reasoning. Dnamic Teaching Tools Dnamic Assessment & Progress Monitoring Tool Interactive Whiteboard Lesson Librar Dnamic Classroom with Dnamic Investigations 7. x 8. x 9 9. x > 7 0. 0 x < 9. 0 x 5. x 0. x > 0. x 0 5. about 0.5 in.. a. about 5.9 ft, about. ft b. about. ft, about 0.5 ft c. no; When the hang time doubles, the height increases b a factor of. 7. (, 0) and (, ); (, ) (, 0) 5 x 8. (, ) and (, ); 8 7 (, ) (, ) 5 x Section 5. Solving Radical Equations and Inequalities 7 9. (0, ) and (, 0); (, 0) 5. (0, ); x (0, ) 5. (, 0), (, ), and (, ) ; (, ) (, 0) x (, ) 50. (, ); 5 x (0, ) (, ) 5 5. See Additional Answers. 5 x Section 5. 7
57. The square root of a quantit cannot be negative. 58. x = 5; The solution of the equation is the x-value of the point of intersection of the graphs. 59. Raising the price would decrease demand. 0. The area of the circle must be at least P square feet. For a circle with radius r, this means πr P. Solving for r gives 57. USING STRUCTURE Explain how ou know the radical equation x + = 5 has no real solution without solving it. 58. HOW DO YOU SEE IT? Use the graph to find the solution of the equation x = x +. Explain our reasoning. = x = x + (5, ) x. MATHEMATICAL CONNECTIONS The Moeraki Boulders along the coast of New Zealand are stone spheres with radii of approximatel feet. A formula for the radius of a sphere is r = S π where S is the surface area of the sphere. Find the surface area of a Moeraki Boulder.. PROBLEM SOLVING You are tring to determine the height of a truncated pramid, which cannot be measured directl. The height h and slant height of the truncated pramid are related b the formula below. r P π.. π. ft. about.9. a. h = h 0 kt πr b. about 5.75 in.. x + x x x + 5. x 5 + x x +. x 5 + x + 5x + x + 5 7. x + x 8 8. g(x) = x x + 0; The graph of g is a reflection in the -axis and a translation units up of the graph of f. 9. g(x) = x x ; The graph of g is a vertical shrink b a factor of followed b a translation units down of the graph of f. 70. g(x) = (x ) + (x ) ; The graph of g is a reflection in the x-axis followed b a translation unit right and units up of the graph of f. 59. WRITING A compan determines that the price p of a product can be modeled b p = 70 0.0x +, where x is the number of units of the product demanded per da. Describe the effect that raising the price has on the number of units demanded. 0. THOUGHT PROVOKING Cit officials rope off a circular area to prepare for a concert in the park. The estimate that each person occupies square feet. Describe how ou can use a radical inequalit to determine the possible radius of the region when P people are expected to attend the concert. Maintaining Mathematical Proficienc Perform the indicated operation. (Section. and Section.) = h + (b b ) In the given formula, b and b are the side lengths of the upper and lower bases of the pramid, respectivel. When = 5, b =, and b =, what is the height of the pramid?. REWRITING A FORMULA A burning candle has a radius of r inches and was initiall h 0 inches tall. After t minutes, the height of the candle has been reduced to h inches. These quantities are related b the formula kt r = π (h 0 h) where k is a constant. Suppose the radius of a candle is 0.875 inch, its initial height is.5 inches, and k = 0.0. a. Rewrite the formula, solving for h in terms of t. b. Use our formula in part (a) to determine the height of the candle after burning 5 minutes. Reviewing what ou learned in previous grades and lessons. (x x + x + ) + (x 7x) 5. (x 5 + x x ) (x 5 ). (x + x + )(x + 5) 7. (x + x + x + x ) (x + ) 5 h Let f(x) = x x +. Write a rule for g. Describe the graph of g as a transformation of the graph of f. (Section.7) 8. g(x) = f( x) + 9. g(x) = f(x) 70. g(x) = f(x ) + Mini-Assessment 8 Chapter 5 Rational Exponents and Radical Functions Solve the equation or inequalit.. x + = x =. x = x 7 The apparent solution x = 5 is extraneous. So, the onl solution is x = 0.. x + + = x The apparent solution x = is extraneous. So, the onl solution is x =.. (x + ) / = x The apparent solution x = is extraneous. So, the onl solution is x =. 5. x x 8 If students need help... Resources b Chapter Practice A and Practice B Puzzle Time Student Journal Practice Differentiating the Lesson Skills Review Handbook If students got it... Resources b Chapter Enrichment and Extension Cumulative Review Start the next Section 8 Chapter 5