MAT116 Final Review Session Chapter 1: Equations, Inequalities, and Modeling

Transcription

1 MAT116 Final Review Session Chapter 1: Equations, Inequalities, and Modeling

2 Solving Equations: Types of Equations in Chapter 1: Linear Absolute Value Quadratic Rational To SOLVE an equation means to find the values for the variable that make the equation a TRUE statement.

3 Examples: Solve each equation. 1. 3(x 5) = 4 2x 2. 4 (x + 2) = 3x x 4 3 = x x 6 = 3 4 x 9

4 Vocabulary An equation that is satisfied by every real number, R, for which both sides are defined is an identity. A conditional equation is an equation that is satisfied by at least one real number but is not an identity. An inconsistent equation is an equation that has no solution. The symbol is used to represent the empty set (the set with no numbers). When an equation has a solution that does not satisfy the original equation the value is called an extraneous solution. ALWAYS check your solutions to an equation in the ORIGINAL problem.

5 Examples: Solve each equation. 5. 4(y 1) = 4y x + 3x = 6x 7. 2x + 3x = 5x + 1

6 Absolute Value Equations 8. x = 7 What values of x would make the statement true? Why? Solve. 9. x 5 = x 7 = x + 9 = x + 3 = 2

7 Quadratic Equations Definition: Quadratic Equation A quadratic equation is an equation of the form ax 2 + bx + c = 0, where a, b, and c are real numbers with a 0. Methods for Solving Quadratics: FACTORING SQUARE ROOT PROPERTY QUADRATIC FORMULA

8 The Zero Factor Property If A and B are algebraic expressions, then the equation AB = 0 is equivalent to A = 0 or B = 0. For Quadratics: If (ax b)(cx d) = 0 then ax - b = 0 or cx - d = 0 The Square Root Property For any real number k, the equation x 2 = k is equivalent to x = ± k x k. The Quadratic Formula The solutions to ax 2 + bx + c = 0, with a 0, are given by the formula: x = b ± b2 4ac 2a

9 Examples: Solve each equation. 13. x 2 x 20 = x 2 5x 3 = x 2 +2 = x 12 3 x = x + 4 x + 7 x + 4 = 2 x + 3 x x 1 9 x +1 = 3 x (3x 1) 2 = x 2 2x = 1

10 Solving Linear Inequalities Solving a linear inequality is the same as solving a linear equation EXCEPT when an inequality is multiplied or divided by a negative number, the direction of the inequality symbol is reversed.

11 Write Solutions in Interval Notation. Inequality Interval Notation x > 4 ( 4, ) x < 4 (, 4 ) x > 4 [ 4, ) x < 4 (, 4 ] x > 4 and x < 7 ( 4, 7 ] x < 4 or x > 7, 4 [ 7, ) All Real Numbers (, )

12 Examples: Solve each inequality. Write your solution set in interval notation x 6 > x x > 7 + x or 4x + 3 > x < 3 2x 11

13 Absolute Value Inequalities Get the Absolute Value by itself on one side Set up 2 inequalities 1st - looks exactly like the problem without the absolute value 2 nd - flip the inequality symbol and the value on the opposite side changes sign (+/-) Between the two inequalities is the word AND or OR GREAT OR gets an OR in between LESS TH AND gets an AND in between ** This is very important. It changes the answer if there is no word or the wrong word between the inequalities.

14 Solve: Write your answer in interval notation x < x x 4

15 The Distance Formula and The Midpoint Formula The Distance Formula The distance d between the points (x 1, y 1 ) and (x 2, y 2 ) is given by the formula d 2 2 x x y y1 The Midpoint Formula The midpoint of the line segment with endpoints (x 1, y 1 ) and (x 2, y 2 ) is x 1 x 2 2, y 1 2 y 2.

16 Example: 28. Find the distance and the midpoint between the points: a. (1, 3) and (4, 7) b. (4, -2) and (-2, 4)

17 Lines Standard Form If A, B, and C are real numbers, then the graph of the equation Ax + By = C is a straight line, provided that A and B are not both zero. Every straight line in the coordinate plane has an equation in the form Ax + By = C, the standard form for the equation of a line. Slope-Intercept Form The equation of the line (in slope-intercept form) with slope m and y-intercept (0, b) is y = mx + b Every non-vertical line has an equation in slope-intercept form. Point-Slope Form The equation of the line (in point-slope form) with slope m and the point (x 1, y 1 ) is y y 1 = m(x x 1 )

18 Information about Lines To find a y-intercept, plug in zero for x and solve for y. This will be where the graph crosses the y-axis. To find an x intercept, plug in zero for y and solve for x. This will be where the graph crosses the x- axis Slope is represented by m. Slope is a ratio written as a fraction: slope m change in change in y - coordinates x - coordinates y x y x 1 rise run

19 Examples: 29. Find the slope of the line: 30. Calculate the slope of the line containing each set of points: (2, -1) and (5, -3)

20 Writing the Equation of a Line You can be given several different pieces of information and be expected to determine the equation of a line. Always remember that there are 2 things that you MUST know: 1. The slope of the line 2. A point on the line You won t always be given these 2 items, but you will have enough information to find them.

21 Example: 31. Write the equation in slope-intercept form of the line passing through the points (-3, 5) and (2, 1).

22 Parallel and Perpendicular Lines Parallel Lines Two non-vertical lines in the coordinate plane are parallel if and only if their slopes are equal. Perpendicular Lines Two lines with slopes m 1 and m 2 are perpendicular if and only if m 1 m 2 = 1. (Opposite sign and reciprocals)

23 Examples: 32. Write the equation in standard form of the line parallel to 4x + 9y = 5 and containing the point ( 4, 2). 33. Write the equation in slope-intercept form of the line perpendicular to 3x y = 9 and containing (0, 0).

24 Applications: 34. (Geometry) Julia s soybean field is 3 m longer than it is wide. To increase her production, she plans to increase both the length and the width by 2 m. If the new field is 46m 2 larger than the old field, then what are the dimensions of the old field?

25 Applications: 35. (Work Rate) Rita can process a batch of insurance claims in 4 hours working alone. Eduardo can process a batch of insurance claims in 2 hours working alone. How long would it take them to process a batch of claims if they worked together?

26 Applications: 36. (Mixture) How many gallons of a 20% alcohol solution must be mixed with 10 gallons of a 50% alcohol solution to obtain a 30% alcohol solution?

27 Applications: 37. (Linear) Speedy printing charges $23 for 200 deluxe business cards and $35 for 500 deluxe business cards. Given that the cost in a linear function of the number of cards printed, find a formula for the function and then find the cost of 700 business cards.

28 Chapter 1 Review Solving Equations Interval Notation Lines Formulas Applications

29 Example Answers 1) x = 11 2) x = 1 2 3) x = 24 4) x = 12 5) Infinitely many solutions 6) x = 0, Conditional equation 7) No solution 8) x = 7 9) x = 8, x = 2 10) x = 7, x = 7 11) No solution 12) x = 1, x = 5 13) x = 5, x = 4 14) x = -1/2, 3 15) x = i 2/3, x = -i 2/3 16) x = 1 2 ± ) x = 1 ± 2 18) x = 8, x = 6 19) x = 2 20) x = 2 21) x > 5, (5, ) 22) x 2 23) (, 3) ( 1, ) 24) [ 4, 5) 25) (0,12) 26) (, 1] [4, ) 27) (, 2] [6, ) 28) a) d = 5, M = (5/2, 5) b) d = 6 2, M = (1, 1) 29) ) ) y = 4 5 x ) y = 4 9 x ) y = 1 3 x 34) w = 9, l = 12 35) 1 hour and 20 minutes 36) x = 20 gallons 37) y = 25x 375, x = $43