Chapter 1 Equations and Inequalities Aim 1 1

Chapter 1 Equations and Inequalities

Aim #1. 1: How do we graph and interpret information? Analytic Geometry this new branch of geometry founded by Rene Descartes brings Algebra and Geometry together. Key Terms Rectangular Coordinate System or Cartesian Coordinate System X-axis, Y-axis Quadrants

Graphs of Equations X 3 Y = 4 – x 2 -5= 4 - 32 (X, Y) (3, -5) Solution of an equation that satisfies the equation.

Using a Graphing Calculator Using your calculator Y = 4 – x 2 Understanding the Viewing Rectangle [-10, 1] Minimum X-value, Maximum X- value and X scale

Intercepts X-intercept- is the point on the graph where it intersects with the X- axis. Ex. (X, 0) Y-intercepts – is the point on the graph where it intersects with the Y-axis. Ex, (0, Y) Look at Text- Identifying Intercepts

Interpreting Information from Graphs Line Graph used to illustrate trends or data over time. Look at Text

Summary: Answer in complete sentences. What is the rectangular coordinate system? Explain why (5, -2) and (-2, 5) do not represent the same point. What does [-20, 2, 1] by [ -4, 5, 0. 5] viewing rectangle mean? Determine whether the following is true or false. Explain. If the product of a point’s coordinates is positive, the point must be in quadrant I.

Warm-up: 8/23/2011 Take out your homework and unit plan. Copy the questions and Answer. Explain how to graph an equation in the rectangular coordinate system. Explain how to graph the point (7, -8) on the rectangular coordinate system. Sketch a function that models the following situation. As the blizzard got worse, the snow fell harder and harder. (Label the x-axis Time and Y-axis snowfall)

Aim #1. 2: How do we solve equations? What is a linear equation? A linear equation has one variable and can be written in the form of y= mx +b, where a and b are real numbers and a ≠ 0. How do we solve? 4 x + 12= 0

Practice: 4 x + 5 = 29 6 x – 3 = 63

How do we solve other types linear equations? Solve 2 (x – 3) – 17 = 13 4( 2 x + 1) = 29 + 3 (2 x – 5)

Practice: 5 x – (2 x – 10 ) = 35 3 x + 5 = 2 x + 13 13 x + 14 = 12 x – 5

How do we solve an equation w/ a fraction? Steps: Multiply the entire equation by a multiple of the denominator. For this example, it would be … This would get rid of all denominators. Then combine like terms and isolate the variable.

Guided Practice:

What is a Rational Equation? It is an equation containing one or more rational expressions. Solving: Hint: Multiply the entire equation by the LCD or multiple of x, 5, and 2 x.

Solving a Rational Equation

Solving a Rational Equation

Categorizing the Different Equations Key Terms 1. Identity Equation is an equation that is true for all values of x. Ex. : x + 3= x + 2 + 1 2. Conditional Equation is true for a particular value of x. Ex. 2 x + 5 = 10 x = 2. 5 3. Inconsistent Equation is an equation that is not true for any value of x. Ex. x = x + 7

Summary: Answer in complete sentences. 1. What is a linear equation in one variable? Give an example. What are some other types of linear equations? 2. Does the following make sense? Explain your reasoning. Although I can solve 3 x + 1/5 = ¼ by first subtracting 1/5 on both sides, I find it easier to multiply by 20, the least common denominator, on both sides. 3. Is the equation (2 x- 3)2 = 25 equivalent to 2 x – 3 = 5? Explain.

Warm-up: 8/25/2011 Take out your homework and unit plan. Copy the questions and Answer. Solve for X. 1. 7 x – 5 = 72 2. 6 x – 3 = 60 3. 13 x + 14 = 12 x – 5

Warm-up: 8/25/2011 Take out your homework and unit plan. Copy the questions and Answer. Solve for X.

Aim #1. 3: How do we use linear equations to model situations? Steps to solving Word Problems 1. Read the problem. Twice. 2. Define x = 3. Write your equation. 4. Solve. 5. Check your solution. Does it make sense?

Using the 5 -step strategy and find the number. When two times a number is decreased by 3, the result is 11. What is the number? Let x= number Equation: 2 x – 3 = 11 Solve for x.

When a number is decreased by 30% of itself, the result is 20. What is the number? Let x = the number Equation x • __ = ___ Solve and Check your answer.

Solving a Formula for a Variable 2 L + 2 W = P Solve for l Your goal is to isolate L. Subtract 2 W on both sides. Then divide by 2 on both sides to isolate L. What’s your final answer? Note: 2 W is like one term because you are multiplying W by 2.

Check for Understanding Solve the formula for W. 2 L + 2 W = P

Solving a Formula for a Variable that Occurs Twice Solve for P. Factor out the P from P + P rt. A=P+Prt Then divide by the expression in parenthesis to isolate P.

Check for Understanding Solve the formula for C. P = C + MC

Summary: Answer in complete sentences. Explain how to solve for P below and then solve. T= D + pm What does it mean to solve for a formula? Write an original word problem that can solved using a linear equation. Then write out all the steps for the solution.

Warm-up: 8/26/2011 Take out your homework and Unit Plan. What is the difference between an identity, conditional and inconsistent equation? Solve for r. Agenda 1. Go over hw. 2. Quiz 3. Review

Warm-up: 8 -29 -2011 Take out your homework and Unit Plan. Solve. Reminder- Quiz Friday- P. 6 B through- 1. 5 Note: We are skipping 1. 4 at this time.

Warm-up: 8/30/2011 Take out your homework and Unit Plan. Solve for x. Reminder Quiz Friday P. 6 B -1. 5

Warm-up: 8 -31 or 9 -1 Take out your homework and unit plan. First, write the value (s) that make the denominator (s) zero. Then solve the equation. What type of equation is this?

Aim #1. 5: How do we solve quadratic equations? Definition of a Quadratic Function: A quadratic equation is an equation that can be written in the general form of ax 2 + bx + c = 0, where a, b, and c are real numbers and a≠ 0. i. e. Quadratics are second degree polynomial functions.

Solving Equations by the Square Root Property Notice in the examples to the left that there is no b term. Steps: 1. Isolate the x 2 term. 2. The find the square root of both sides. 3. Simplify.

Check for Understanding: Solve by the Square Root Property.

Zero-Product Principle If the product of two algebraic expressions is zero, then at least one of the factors must equal to zero. If AB= 0 then A= 0 or B= 0 Example: x 2 + 7 x + 10 = 0 1. Factor. 2. Set each factor = 0 3. Solve. (X + 5)(x+2) = 0 X+ 5 = 0 or x+2 = 0 x = -5 or x = -2

Solving Quadratic Equations by Factoring a. Is there a GCF, that can factored out? b. Subtract 4 on both sides and set equation = 0. Now factor.

Solving Quadratics by Completing the Square is a strategy to solve quadratics when: 1. The Trinomial can not be factored 2. Zero Product property can not be used Completing the square allows us to convert the equation so that it can be solved using the square root property.

Completing the Square: If x + bx is a binomial, then by adding , which is the square of half the coefficient of x, a perfect square trinomial will result. That is,

Completing the Square What term should be added to each binomial so that it becomes a perfect square trinomial? Write and factor the trinomial. a. x 2+ 8 x b. Solution:

Completing the Square What term should be added to each binomial so that it becomes a perfect square trinomial? Write and factor the trinomial.

Completing the Square What term should be added to each binomial so that it becomes a perfect square trinomial? Write and factor the trinomial.

Solving Quadratics using Completing the Square Steps: 1. Subtract 4 on both sides. 2. Take the b term and divide by 2 and square it. 3. Now add it to both sides of the equation. 4. Now you can express the left hand side as a square. 5. Apply the square root property and solve for x.

Solving Quadratics using Completing the Square Guided Practice:

Solving Quadratics using Completing the Square Steps: 1. Divide the entire equation by 9, so that a = 1. 2. Add -4/9 to both sides. 3. Complete the square. 4. Then solve for x.

Solving Quadratics using Completing the Square Guided Practice:

What is the Quadratic Formula? Quadratic formula: If ax 2 + bx + c = 0 and a≠ 0

Using the Quadratic Formula Solve x 2 + 6 = 5 x Steps: 1. Subtract 5 x on both sides, so the equation = 0. 2. Identify the values for a, b, and c. a= 1, b = -5, c = 6 3. Then substitute into the

What is the discriminant? Property of the Discriminant For the equation ax 2 + bx + c= 0, where a ǂ 0, you can use the value of the discriminant to determine that number of solutions. If b 2 – 4 ac > 0, there are two solutions. If b 2 – 4 ac = 0, there is one solution. If b 2 – 4 ac <0, there are no solutions

Using the Discriminant For each equation, compute the discriminant. Then determine the number and type of solutions.

Summary: Answer the following in complete sentences. 1. What is a quadratic equation? 2. What are at least 3 different ways of solving a quadratic? 3. When is using the square root property helpful? (Think of at least 2 ways. ) 4. Solve using any method. Explain the method you choose and why.

Warm-up: 9 -2 -2011 Take out your homework and Unit Plan. Find the x intercepts of the equation. (i. e. Solve. ) BE READY, YOU WILL BE CALLED TO THE BOARD IN 10 MINUTES.

Warm-up: 9 -6 -011 Take out your homework and Unit Plan. I will collect Chapter Review with work on TEST DAY!!! Find the x-intercepts. BE READY, YOU WILL BE CALLED TO THE BOARD IN 10 MINUTES.

Warm-up: 9 -9 -2011 For the following equation, compute the discriminant. Then determine the number of solutions and type of solutions: BE READY, YOU WILL BE CALLED TO THE BOARD IN 8 MINUTES.

Aim#1. 5 B: How do we solve problems modeled by quadratic equations? In a 25 -inch television set, the length of the screen’s diagonal is 25 inches. If the screen’s height is 15 inches, what is its width? You need to use the Pythagorean Theorem: a 2 + b 2 = c 2 Sketch a figure. Substitute into the equation what you know and Solve. Write your final answer in a sentence.

Problem: What is the width of a 15 -inch television set whose height is 9 inches?

Summary: Answer in complete sentences. If you are given a quadratic equation, how do you determine which method to use to solve it? Describe the relationship between the solutions of: ax 2 + bx + c =0 and the graph of y = ax 2 + bx + c. Write a quadratic equation in general form whose solution set is

Warm-up: 9 -12 -2011 Take out your homework. - Do not copy problem. Show all work and solve. Define your variables, write your equation, solve and express final answer as a sentence. After a 20% reduction, you purchase a television for $336. What was the television’s price before the reduction? Including 5% sales tax, an inn charges $252 per night. Find the inn’s nightly cost before the tax is added. Each side of a square is lengthened by 2 inches. The area of this new, larger square is 36 sq. inches. Find the length of a side of the original square.

Aim # 1. 6: How do we solve polynomial equations by factoring? A polynomial equation is the result of setting two polynomials equal to each other. The equation is in general form if one side is 0. and the polynomial on the other side is in descending powers of the variable. The degree of a polynomial is the same as the degree of any term in the equation. Here are some examples of polynomial equation: 3 x + 5 = 14 Degree of 1 also linear 2 x 2+7 x =4 Degree of 2 also quadratic x 3 + x 2 = 4 x + 4 Degree of 3 also cubic

Solving a Polynomial by Factoring Steps: 1. Move all terms to one side and set equation = 0. Subtract 27 x 2 on both sides. 2. Factor the GCF. 3. Set each factor to 0 4. Solve.

Guided Practice: Solve by factoring: 4 x 4=12 x 2

Solving a Polynomial Equation x 3 + x 2 = 4 x + 4 Steps: 1. Move all terms to one side and set equation = 0. 2. Factor using the grouping strategy. 3. Set each factor to zero and solve.

Guided Practice: Solve by factoring: 2 x 3 + 3 x 2 = 8 x + 12

Practice: Solve each polynomial equation by factoring and then use zero-product principle. 1. 5 x 4 - 20 x 2 = 0 2. 4 x 3 -12 x 2 = 9 x – 27 3. x + 1 = 9 x 3 + 9 x 2 4. 9 y 3 + 8 = 4 y + 18 y 2

How do we solve radical equations? A radical equation is an equation in which the variable occurs in a square root, cube root or any higher root. An example is: Squaring both sides. Eliminates radical sign.

Note: We solve radical equations with nth roots by raising both sides of the equation to the nth power. Unfortunately, if n is even, all the solutions of the equation raised to the even power may not be solutions of the original equation. For example: x=4 If we square both sides, we obtain x 2= 16 x = + √ 16 = + 4 This equation has two new solutions, -4 and 4. By contrast only 4 is a solution to the original equation.

When raising both sides of an equation to an even power, always check proposed solutions in the original equation.

Solving Radical Equations Containing nth Roots 1. If necessary, arrange terms so that one with the radical is isolated on one side of the equation. 2. Raise both sides of the equation to the nth power to eliminate the nth root 3. Solve the resulting equation. If there is still a radical repeat step 1 and 2. 4. Check all proposed solutions.

Extraneous solutions or extraneous roots are solutions that do not satisfy the original equation.

Solving a Radical Equation Steps: 1. Isolate radical on one side by subtracting 2 on both sides. 2. Raise both sides to the nth power. Because n, the index is 2, we square both sides. 3. Solve the resulting equation.

Guided Practice: Solve:

How do we solving an equation that has two radicals? Steps: 1. Isolate a radical on one side. 2. Square both sides. 3. Simplify. 4. Note the resulting equation still has a radical sign so, repeat steps 1 and 2. 5. Solve resulting equation.

Guided Practice:

Practice: Solve each radical equation. Check all proposed solutions.

Summary: Answer in complete sentences. Without actually solving the equation, give a general description of how to solve x 3 - 5 x 2 – x + 5 =0. In solving why is it a good idea to isolate a radical term? What if we don’t do this and simply square each side? Describe what happens. What is an extraneous solution to a radical equation?

Warm-up: 9 -13 -11 Take out your homework and Unit Plan. Solve each polynomial equation by factoring. Then use the zero product principle. 1. 3 x 3 +2 x 2 = 12 x + 8 2. 2 x 4 = 16 x 3. 2 x 3 – x 2 – 18 x + 9 = 0 BE READY, YOU WILL BE CALLED TO THE BOARD IN 6 MINUTES. Reminder: Quiz & Warm-up due Friday!

Warm-up: 9 -14 -2011 Take out your homework and Unit Plan. Solve each radical equation. Check all proposed solutions.

Aim #1. 6 B: How do we solve equations with rational exponents? We know that expressions with rational expressions represent radicals:

SOLVING RATIONAL EQUATIONS OF THE FORM Assume that m and n are positive integers, m/n is in lowest terms, and k is a real number. 1. Isolate the expression with the rational exponent. 2. Raise both sides of the equation to the n/m power. If m is even: If m is odd:

NOTE: It is incorrect to insert + symbol when the numerator of the exponent is odd. An odd index has only ONE root. 3. Check all proposed solutions in the original equation to find out if they are actual solutions or extraneous solutions.

Solving Equations Involving Rational Exponents Solve: Steps: Goal is to isolate the expression with the rational exponent. Undo the addition or subtraction. Undo the Multiplication or division. Raise both sides by the reciprocal of the exponent. Note: exponent is odd so we do not add +. Check proposed solution.

Guided Practice: Solve:

Practice:

How do we solve equations that are in quadratic form? An equation that is quadratic in form is one that can be expressed using an appropriate substitution.

Solving an Equation in Quadratic Form Steps: 1. Replace x 2 with u. 2. Factor. 3. Apply the zero-product principle. 4. Solve for u. 5. Then replace u with x 2 and now solve for x.

Guided Practice: Solve:

Solving an Equation in Quadratic form Steps: 1. Replace 2. Factor. 3. Set each factor to 0. 4. Solve for u. with u.

How do we solve equations with Absolute Value? An absolute value is the distance a number is from zero. An absolute value equation: What values of x would make the above equation true?

Solving Absolute Value Equations Solve: Steps: Rewrite equation without absolute value bars. Write one equation = 11 and a second = -11 Solve each equation. Check all proposed solutions.

Solving Absolute Value Equations NOTE: Before we can solve we need to isolate the absolute value expression.

Practice: Solve:

Summary: Answer in complete sentences. Explain in words how to solve. Then show the work. How do we solve an absolute value equation? Include an example to support your answer. What do solving absolute value equations and radical equations have in common? What other type of equation did we learn to solve in this Aim 1 B? Give an example.

Warm-up: 9/ 16/2011 Take out your homework. Make the appropriate substitution, then solve.

Warm-up: 9 -19 -2011 Solve each equation. Make the appropriate substitution. Be ready, in 8 minutes you will be called to the board.

Aim #1. 7: What is interval notation and how do we use it? Goals of this Aim are: Use of Interval Notation Find Intersections and Unions of Intervals Solve Linear Inequalities

What is interval notation? Subsets of real numbers can be represented using interval notation. Examples: a. Open Interval: (a, b) represents the set between the real numbers a and b and not including a and b. (a, b) = {x /a < x < b} b. Closed Interval: [a, b] represents the set between the real numbers a and b including a and b. [a, b] = {x /a < x < b}

More Examples of Infinite Notation The infinite interval represents the set of real numbers that are greater than a. The infinite interval represents the set of real numbers that are less than or equal to b. Remember: Parentheses indicate endpoints not included in an interval. Square brackets indicate that endpoints that are included interval

Using Interval Notation Express each interval using set builder notation and graph. a. (-1, 4] = {x /-1< x < 4} b. (2. 5, 4] = c. (-4,

How do we find the intersection and union of intervals? Use graphs to find each set: Steps: 1. Graph each interval on a number line. 2 a. To find intersection, take the portion of the number line that the two graphs have in common. b. To find the union, take the portion of the number line representing the total collection of numbers in the two graphs.

Practice: Use graphs to find each set:

How do we solve Linear Inequalities in One Variable? Solve and graph 3 - 2 x < 11. Try: 2 -3 x < 5

Solving a Linear Inequality Solve and graph. -2 x – 4 > x + 5 3 x + 1 > 7 x - 15

How do we recognize inequalities with unusual solution sets? Examples: x < x + 1 The solution is all real numbers. Or using interval notation its___. If you attempt to solve an inequality that has no solution sets, you will eliminate the variable and obtain a false statement such as 0 > 1. If you attempts to solve an inequality that is true for all real numbers, you will eliminate the variable and obtain a true statement such as 0 < 1.

Solving Linear Inequalities Solve each inequality. 2 (x + 4)> 2 x + 3 x + 7 < x - 2

Summary: Answer in complete sentences. When graphing the solutions of an inequality, what does a parenthesis signify? What does a bracket signify? Describe the ways that solving a linear inequality is similar to solving a linear equation. Describe ways that solving a linear inequality is different from solving a linear equation.

Warm-up: 9 -20 -2011 Take out your homework and Unit Plan. Solve: Solve. Express answer in interval notation and graph. C. 18 x + 45 < 12 x - 8 d. -4 (x + 2) > 3 x + 20

Warm-up: 9 -21 or 9 -22 Take out your homework and Unit Plan. Solve each linear inequality. Be READY you will be CALLED in 8 minutes to the board.

Aim #1. 7 B: How do we solve other types of inequalities? Two inequalities, such as : -3 < 2 x + 1 and 2 x + 1 < 3 That can written as a compound inequality. -3< 2 x + 1 < 3

How do we Solve a Compound Inequality? Solve and graph. -3 < 2 x + 1 < 3 Goal is to isolate variable. Subtract from all parts. Simplify. Divide each part by 2. Simplify. Now you try: 1< 2 x + 3 < 11

How do we Solve Inequalities with Absolute Value? Solving an Absolute Value Inequality If X is an algebraic expression and c is a positive number. The solutions of are the numbers that satisfy –c< X < c. X < -c or X > c. These rules are valid if < replaced by <, And > is replaced by >.

Solving an Absolute Value Inequality Given the inequality: Solve the compound inequality. The solution set written in interval notation is: And the graph is …

Solving an Absolute Value Inequality Solve and graph the solution on a number line:

Solving an Absolute Value Inequality Steps: Remember to isolate Absolute Value Expression Then before you can rewrite without bars

Solving an Absolute Value Inequality Solve and graph on a number line:

Solving an Absolute Value Inequality Solve and graph the solution set on a number line:

Solve and graph on a number line:

Applications: Acme car rental agency charges $ 4 a day plus $0. 15 per mile. Interstate rental agency charges $20 a day and $0. 05 per mile. How many miles must be driven to make the daily cost of an Acme rental a better deal than an Interstate rental? First define your variable: Let x = Represent both quantities in terms of x. Which inequality symbol do we use and why? Write and solve the inequality. Check your proposed solution.

Practice: A car can be rented from Basic Rental for $260 per week with no extra charge for mileage. Continental charges $80 per week plus 25 cents for each mile driven to rent the same car. How many miles must be driven in a week to make this rental cost for Basic Rental a better deal than Continental’s?

Summary: Answer in complete sentences. Describe how to solve an absolute value inequality involving this symbol <. Give an example. Describe the solution set of What’s wrong with this argument? Suppose x and y represent two real numbers, where x > y:

Warm-up: 9 -23 -2011 Take out your homework and Unit Plan. Solve each inequality.

Please Read DO NOW: 9 -26 -2011 Take out your Review, Work and Unit Plan. Turn in last week’s warm-up in the tray on table. Complete Today’s Warm-up in your notebook. Reminder: Crossword and Unit Plan due Tuesday, Sept. 27 th Test Thursday