# Equations and Inequalities

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1 Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations. Equations and Inequalities Main Overarching Questions: 1. How do you solve rational equations?. What is an extraneous solution? Objectives: Activities and Questions to ask students: Solve rational equations Determine whether an equation represents an identity, conditional, or inconsistent statement. Give students a simple example of a rational equation: 1 3 x x. Ask students to observe what value of x is not allowed. If they are not sure, says we are never allowed to have blank in the denominator. Next ask them how they might proceed to solve the equation. Why is this equation NOT linear? (It has fractions) How can we clear our fractions? What one quantity can we multiply by to ensure all fractions are cleared? Ask students what the LCD is in the previous example 1 3 x x. Do we have to multiply the by the LCD? Why? Have students multiply through by LCD to clear the fraction. What kind of equation is left over? How do we solve it? Have students summarize the process of solving radical equations. x 3 Give students another rational equation: 9. Ask them to solve it. x 3 x 3 What solution did you get? Does this solution work when plugged in? How could we have known the solution would not work? Define the solution x = 3 in the previous example as an extraneous solution. Remind students to always list out restrictions on the variable at the beginning of the problem. Since this is a MAC 1105 topic, make sure to graduate to more difficult problems that involve multiple factors in the LCD. Give students a simple identity: x x for example. Ask them what they notice? What is the solution to this equation? 0/01/10 1

11 Solve equations using substitutions. Solve absolute value equations. Now, how about 1 x 3? What do we need to do to both sides? How are 3 and 1 3 related? How are and 1 related? Now, have students try x 3 4. They should see what to do, but they may have trouble evaluating x 4 3. Have them use the definition of rational exponents to calculate this value and use the calculator as a check. Was 8 the only solution to the previous equation? Would -8 have also worked? Have students show by substitution that both 8 and -8 are solutions. Repeat with another rational exponent with an even numerator. Now, follow the same steps with an exponent with an odd numerator. Do both the positive and negative solutions work? Have students draw the conclusion that if the power has an even numerator, we keep both the positive and negative solutions. If the power has an odd numerator, we only take the positive solution. Have students work an example, where they have to isolate the rational exponent term first. After solving, have them summarize their steps. 4 Give students a simple equation in quadratic form like: x 5x 4 0. What do they notice? Have them compare it with the quadratic equation x 5x 4 0. Since we know how to solve the latter equation, can we somehow make a substitution to make the former equation look like the latter? It will most likely be difficult for the students to see this one on their own. Have students think about it for a minute, then suggest (if no one else has) to let x u. Tell students that we want to write the equation in terms of u now. 4 What is x then? Have students transform the equation to u 5u 4 0. Temporarily we should forget about the original equation and just focus on solving the u equation. How do we solve this? What type of equation is it? Have students solve for u. Then ask, are we finished? Did we solve the original equation? NO! We needed x not u. How do we get x? What is the relationship between u and x? The substitution linked the two variables. Have students substitute each value of u to find x. 6 3 Give students another example to try, but give different power of x like x and x. Ask students if there is a pattern to what u is. What is absolute value? Can more than one number have the same absolute value? Give an example and explain your reasoning. What number is the exception? For x = 5, what numbers could replace the x? How many solutions are there? Write 0/01/10 11

12 equations for these solutions. For x + 1 = 5, what two numbers could x be? Using our equations for the example above, write two equations to solve this problem. Write an absolute value equation where you might have only one solution. Write an absolute value equation where you might have no solution. Now give the students an equation where the absolute value is not isolated. What needs to be done first to solve x = 9? Solve and check. Now, if x = what must be true about x? Must x =? Write equations for x. If x = or x = -, then how can we solve an equation like x+ = x 3 using opposites? Ask students to solve and check both answers. Systems of Equations Overview of Objectives, students should be able to: 1. Determine if a given ordered pair is a solution to a system of linear equations.. Solve systems of linear equations using graphing. 3. Solve systems of linear equations by substitution. 4. Solve systems of linear equations by addition. 5. Identify systems that have no solution or infinitely many solutions. 6. Solve systems of linear equations in three variables. Objectives: Determine if a given ordered pair is a solution to a system of linear equations. Solve systems of linear equations using graphing Main Overarching Questions: 1. When is an ordered pair a solution of a system?. How do you solve systems of equations by graphing? 3. How do you solve systems of equations by algebraic methods? 4. Compare and contrast the methods of solving systems for efficiency and accuracy. 5. How do you determine when a system has no solution or infinitely many solutions? Activities and Questions to ask students: Present two linear equations and have students substitute an ordered pair into x and y. What is meant by the term solution to an equation? Does the ordered pair create true or false statements? Is this point a solution for both of these equations? When is an ordered a solution of a system of linear equations? Students will graph 3 systems of linear equations: a pair of intersecting lines, a pair of parallel lines, and two equations that are the same line. Direct students to compare the three systems: describe the type of lines, describe how many points they have in common, and compare the equations within each system Students may present their results. 0/01/10 1

14 Solve systems of linear equations in three variables. and the number of solutions. Students should see this is an extension of solving systems of linear equations in two variables. Begin by giving students a system of linear equations in three variables and a numeric ordered triple in the form ( x, y, z). How do the numbers and variables correspond? How can we verify the ordered triple satisfies the system? Ask students to think about how we solved systems in two variables. Is there a way we can transform the three variable system to a two variable system? To facilitate the discussion it would help to give a simple linear system in three variables, where one variable is easily cleared. Some students will suggest eliminating one variable using two equations. However, ask them how many equations are needed to solve a system of two variables. How can we get another equation in terms of the two remaining variables? Have students conclude they need to eliminate one variable from two PAIRS of equations resulting in two equations in terms of the two remaining variables. Now that we have two equations in terms of two variables, how do we solve? If students have trouble, have them discuss the ways we solved systems of equations earlier. Students should now be able to solve and get numeric values for the two variables. How do we get the value of the third variable we eliminated before? Ask students to specific about which equation they use to find the third variable. Does it matter? Will the answer be different? Have students check their ordered triple in the original system. Solving Linear and Absolute Value Inequalities Overview of Objectives, students should be able to: 1. Use interval notation to represent solutions to inequalities in one variable.. Find intersections and unions of intervals. 3. Solve linear inequalities in one variable. 4. Recognize linear inequalities that have no solution or infinitely many solutions. 5. Solve compound inequalities. 6. Solve absolute value inequalities. Main Overarching Questions: 1. How do you solve and graph inequalities in one variable?. How do you determine the number of solutions to a linear inequality or if no solution exists? 3. How do you use set notation and interval notation to express the solutions of linear inequalities? Objectives: Activities and Questions to ask students: Use interval notation to represent solutions to If x 3, what number(s) does this x stand for? What is the least number included? Is 3 0/01/10 14

15 inequalities in one variable. included? Why or why not? We use (3, ) to show all real numbers great than 3. How can we show all real numbers > 7? > -? We use (,4) to show all real numbers less than 4. How do you use interval notation to show all real numbers < 8? < -3? If x is greater than OR EQUAL TO 5, then using interval notation we write [5, ) and if x is less than OR EQUAL TO, we write (,]. How do you express all real numbers less than or equal to 7 in interval notation? Find intersections and unions of intervals. Begin by giving two intervals that overlap over some interval. Describe as the intersection or overlap of two sets. How can we tell where the two sets overlap? Is there a visual way we can accomplish this? Describe as the union or total collection of both intervals. Have students practice finding unions of two intervals. Solve linear inequalities in one variable. If necessary, review properties used in solving equations. How would we solve x 6? How can we use these properties to solve x 6 How is the solution to an inequality different from a solution to an equation? How can we check our solutions? Ask students to solve 3x 9and check their solution. If students fail to change < to >, ask why their solution does not work when checked? Or ask students to divide or multiply both sides of a true inequality like < 4 by -1. Is this still a true inequality? Why or why not? What must be done to make the solution of 3x 9work? After what other operation will you need to switch the inequality sign? Give students more involved problems to work, including a linear equality that contains fractions. Recognize linear inequalities that have no Can you think of a number for x so that x x 5 is a true statement? Try a positive, a solution or infinitely many solutions. negative, and zero. Will x 5 always be greater than x? How many solutions will this inequality have? If we did not see this was a special type of inequality, what would have happened if we solved the inequality for x? 0 5, which is always a FALSE statement. Can you think of a number for x so that x x 5 is a true statement? Can you think of more numbers? How many solutions will this inequality have? If we did not see this was a special type of inequality, what would have happened if we solved the inequality for x? 0 5, which is always a TRUE statement. Solve compound inequalities. Recall that and statements and between statements are the same. If5 x 8, what numbers could x represent? How could we write this is as two inequalities? Now that students understand this type of inequality, we need to solve them. Give a simple 0/01/10 15

17 Graph (solve) a system of linear equalities in two variables. Graph the line again on a new grid and ask students to name a point that makes y x 1. Shade the side where the point lies. Define boundary line and the difference between dashed and solid boundary lines. What is the difference in the solution to y x 1and y x 1? Background knowledge. Review graphing a system of linear equations. What is the solution to a system with intersecting lines? Ask students to graph two inequalities that intersect on the same grid. What points do the two graphs have in common? How many? How does the graph show this? Ask student to graph two inequalities on the same grid that do not overlap. What points do the inequalities have in common? What is the solution to this system? Problem Solving and Modeling Overview of Objectives, students should be able to: 1. Use linear equations to solve problems. Solve a formula for a variable. Objectives: Use linear equations to solve problems Main Overarching Questions: 1. How do you setup and solve a linear equation to solve a problem?. How do you solve for a variable in a formula? Activities and Questions to ask students: In these types of sections, it s best to begin with a word problem and discuss solutions with the students. For example consider this problem: A new car is worth \$4000 but depreciates by \$3000 per year. First we want to determine a model for the worth of the car after x number of years. How much is the car worth in year 0? What about at the end of the first year? nd year? What s the pattern? Can you write an equation to describe the worth of the car after x years? Discuss general problem solving strategies with students. Solve a formula for a variable. Give an example of a formula like E mc. What is different about this equation? How many variables are there? Next ask the students to solve the formula for m. What do you need to move? How do you move it? Can you get a numeric value for m? Why not? Have students practice solving for a variable in a given formula. The contents of this website were developed under Congressionally-directed grants (P116Z090305) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government. 0/01/10 17

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