Chapter 3 September 11, 2017 3.1 Solving equations Solving Linear Equations: These are equations that can be written as ax + b = 0. Move all the variables to one side of the equation and all the constants to the other side. Divide by the coefficient of x. Example 1. Solve the equation 4x + 7 = 3x + 13 for x. Solving Quadratic Equations A quadratic equation is an equation that can be written as ax 2 + bx + c = 0, referred to as the standard form. There are three methods to solve quadratics: Factoring, Complete the Square, and Using the Quadratic Formula. 1. Solving by Factoring Theorem. If ab = 0, then a = 0 or b = 0 or both. Strategy: (1). Write the equation in standard form: ax 2 + bx + c = 0. (2). Factor (3). Set each factor equal to zero and solve. Example 2. Solve the following equations: (a) x 2 + 5x 3 = 9 x 2 (b) 5x 2 3x 30 = x 2 3x + 6 1
2. Solving by Completing Squares Example: (x 3) 2 = 7 Method for Completing the Square: (1). Write the equation in the form: x 2 + dx = e. (move the constant to the right side, divide all terms by the coefficient of x 2 ) (2). Add ( d 2 )2 to both sides. (3). Now the left hand side should factor as (x + d 2 )2. (4). Solve the equation. Example. Solve the following equations by completing the square: (a) x 2 + 5x + 3 = 0 (b) 5x 2 + 7x 2 = 0 2
3. Using the Quadratic Formula Theorem. The solution to any quadratic equation of the form ax 2 + bx + c = 0 is given by x = b ± b 2 4ac 2a To use this formula, your equation must be in standard form ax 2 + bx + c = 0! Example. Use the quadratic formula to solve 2x 2 + 8x + 5 = 0. Example. Solve the equation 5x 2 = x 1. 3
Equations in Quadratic Form: An equation is in quadratic form when it can be arranged to look like a( ) 2 + b( ) + c = 0 where you have the same expression in both the blanks above. So its a quadratic equation where the x has been replaced by a more complicated expression. To solve an expression like this do the following: 1. Replace the more complicated expression with u. 2. Solve the quadratic equation au 2 + bu + c = 0. 3. Set the expression you replaced with u equal to the solutions from 2 and solve. Example Solve the following. (a) x 4 4x 2 13 = 0. (b) y 2 3 y 1 3 56 = 0 4
Rational Equations: A rational equation is an equation with rational expressions. To solve, eliminate the fractions by multiplying both sides by the least common denominator. Note. Be careful! Solving these equations will sometimes produce an extraneous solution, which does not solve the original equation and is not really a solution. Example. Solve the following equations. (a) 6 x 5 2 x = 1 (b) 2 + 5 x 4 = x + 1 x 4 5
Radical Equations: To solve an equation that contains one or more radicals do the following: 1. Isolate one radical on one side of the equation. 2. Raise both sides of the equation to the appropriate power to remove the radical (for example, square roots should be squared) 3. If necessary, repeat the process until all radicals have been removed, then solve the equation. 4. This method also produces extraneous solutions. You MUST CHECK your solutions to make sure they are actually solutions. Example. Solve the following equations. (a) 2 x 3 + 4 = 11 (b) x + 5 + 1 = x (c) 2 x x 3 = 3 6
Absolute Value Equations: Recall, that { x if x 0 x = x if x < 0 Therefore, for the absolute value of an expression to equal a number, then the expression must equal the number or must equal the negative of the number. Then to solve the equations we rewrite every absolute value as two separate equations. Note. Solving absolute value equations can produce extraneous solutions. You MUST CHECK your solutions to make sure they are actually solutions. Example Solve the equations. (a) 2 x = 7 (b) x 5 = 10 (c) x 2 8 = 2x A geometrical interpretation of absolute value equations: Recall that a b refers to the distance between a and b on the number line. Therefore, the equation x 5 = 10 is finding numbers that are a distance of 10 from 5. 7
Equations in Several Variables: When an equation has several variables, we often solve for one variable in terms of the others, i.e., solve for one variable and get an expression that contains the other variables. Examples. (a) The height of a ball in feet thrown with an initivial velocity v in feet per second with initial height h in feet is given by s = 16t 2 vt + h. Solve this equation for v. (b) The formula for the surface area of a right circular cone is given by S = πr(l + r). Solve the equation for r. 8
3.2 Solving Inequalities Definition. An inequality is a statement comparing two quantities which may not be the same: a is less than b: a is less than or equal to b: a is greater than b: a is greater than or equal to b: a < b a b a > b a b Interval Notation: We want to write our answers in interval notation. In this notation, (a, b) means all the numbers between a and b. The parentheses, ( and ), means to exclude then endpoints a and b. Brackets, [ or ], mean to include the endpoint. Here are several examples: Interval Meaning (-3, 8) all the numbers between -3 and 8, not equal -3, 8 (7, 14] all the numbers between 7 and 14, and including 14, but not 7 (, ) all the real numbers [ 3, ) all the numbers greater than -3 and including -3 ( 2, 3) (7, ) all the numbers between -2 and 3 and all the numbers greater than 7, but not 2, 3, or 7 Note. The symbol means to include both intervals in the set. Linear Inequalites: To solve a linear inequality, isolate the variable just like solving a linear equation. Remember: If you multiply or divide an inequality by a negative sign, you must switch the inequality symbol: change < to >, to, or vice versa. Example. Solve the following inequalities. Write the answer in interval notation. (a) 8 3x 12 (b) 7 2 + 4 3 x 5 Number line: We can also represent an inequality as a graph on the number line. The differently colored line represents the solution set, and filled and unfilled circles indicate the the endpoint is included (filled) or excluded (filled). 9
Definition. The statement a < x < b means that x is between a and b, but not including a or b, i.e., a < x and x < b. To solve these combined inequalities, just split them into the two simpler inequalities and solve. Example. Solve the following inequalities. Write the answer in interval notation. Also, graph the solution on the number line. (a) 7 5 4x < 12 (b) 5x 3 < 6x 8 < 4x 5 (c) 5 2x < 3x + 1 < 5x 2 10
Absolute Value Inequalities: Inequalities with an absolute value can be thought of as a distance. Example. Draw a graph on the number line for the following two inequalities. (a) x < 4 (b) x > 4 Solving Absolute Value Inequalities: We need to break the absolute value into two different inequalities. These are the different cases: x < a is equivalent to a < x < a. x a is equivalent to -a x a. x > a is equivalent to x > a OR x < a x a is equivalent to x a OR x a Example. Solve the following inequalities and write your answer in interval notation. (a) 2x 3 < 5 (b) 4x 5 + 5 < 3 (c) 4x + 7 6 9 (d) x 5 > 8 (e) 5x 4 5 13 11
Nonlinear Inequalities Solution Method: 1. Move every term of the inequality to one side. 2. Find the critical values, which are the values for which the expression is equal to zero or undefined. 3. Use the critical values to divide the real number line into intervals 4. Each interval is either entirely in the solution set or entirely out of the solution set. Either test a single number (not an endpoint!) or use a sign chart to determine whether each interval is a solution or not. 5. Determine whether any of the critical values (endpoints of the intervals) are in the solution set. Example. Solve the inequalities and write the answers in interval notations. (a) x 2 + 4x 5 0 (b) x 4 2x 3 > 8x 2 (c) 2x 7 x 5 3 (d) 4x2 16 x 2 5x 3 12