Unit 11: Linear Equations and Inequalities
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- Margaret Drusilla Crawford
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1 Section 11.1: General Form ax + by = c Section 11.2: Applications General Form Section 11.3: Linear Inequalities in Two Variables Section 11.4: Graphing Linear Inequalities in Two Variables KEY TERMS AND CONCEPTS Look for the following terms and concepts as you work through the Media Lesson. In the space below, explain the meaning of each of these concepts and terms in your own words. Provide examples that are not identical to those in the Media Lesson. General (Standard) Form How to Graph a Linear Equation given in General Form Linear Inequality in Two Variables
2 Solution Set to a Linear Inequality in Two Variables The graph of a Linear Inequality in Two Variables How to graph the Solution Set of a Linear Inequality in Two Variables
3 Unit 11: Media Lesson Section 11.1: General Form: ax + by = c Slope-Intercept Form of a Linear Equation y = mx + b x = input, y = output m = slope b = vertical intercept (0, b) General (Standard) Form of a Linear Equation ax + by = c x = input, y = output a, b, and c are constants Example 1: Consider the linear equation 3x 5y = 30 a. Write this equation in slope-intercept form. b. Identify the slope. Determining Intercepts: To find the vertical intercept, set x = 0 and solve for y. To find the horizontal intercept, set y = 0 and solve for x.
4 Media Lesson c. Determine the vertical intercept. d. Determine the horizontal intercept. Example 2: Draw an accurate graph of the function 3x + 2y = 16. Slope-Intercept Form: Slope: Vertical Intercept: Horizontal Intercept: Additional points on the line:
5 Media Lesson Section 11.1 YOU TRY Draw an accurate graph of the function 4x y = 7 Slope-Intercept Form: Slope: Vertical Intercept: Horizontal Intercept: Additional points on the line:
6 Media Lesson Section 11.2: Applications General Form Example 1: Movie tickets cost $7 for adults (matinee), $5.50 for children. A total of $668 was collected in ticket sales for the Saturday matinee. a. Write an equation representing the total amount of money collected. b. If 42 adult tickets were purchased for this matinee, how many children were there? Section 11.2 YOU TRY At a concession stand, two hot dogs and three sodas cost $12. a. Let h represent the price of each hot dog, and s represent the price of each soda. Write a linear equation in general form to represent this situation. b. If sodas cost $1.50 each, how much is each hot dog?
7 Media Lesson Section 11.3: Linear Inequalities in Two Variables Example 1: Graph the equation y = 2x 3 Example 2: Which of the ordered pairs below satisfy the equation y 2x 3? (5, 3) (2, 1) (0, 0) Example 3: Which of the ordered pairs below satisfy the inequality y 2x 3? (5, 3) (2, 1) (0, 0) Example 4: Graph the linear inequality y 2x 3
8 Media Lesson. Example 5: Which of the ordered pairs below satisfy the inequality? y 2x 3 (5, 3) (2, 1) (0, 0) Example 6: Graph the linear inequality y 2x 3 Section 11.3 You Try Complete the problems below. Show as much work as possible, as demonstrated in the Media Examples. a. Which of the ordered pairs below satisfy the linear inequality y 4 2x? (1,2) (0, 0) (5, 0) b. Which of the ordered pairs below satisfy the linear inequality y < 4 2x? (1,2) (0, 0) (5, 0)
9 Media Lesson Section 11.4: Graphing Linear Inequalities in Two Variables Graphing The Solution Set of a Linear Inequality in Two Variables Step 1: Rewrite the inequality as an equality statement. Step 2: Graph the linear equation. This is the boundary of the solution region. Step 3: Determine if the line should be solid or dotted. If the original inequality statement is either < or >, draw a dotted line. If the original inequality statement is either or, draw a solid line. Step 4: Choose a test point and plug it into the original inequality. If the test point satisfies the inequality, shade in the direction of the test point. If the test point does not satisfy the inequality, shade in the opposite direction of the test point. Example 1: Graph the inequality y < 5 3x Example 2: Graph the inequality 3x 2y 6
10 Media Lesson Example 3: Graph the inequality y 2x Section 11.4 You Try Graph the inequality y > 2x 1
11 Unit 11: Practice Problems Skills Practice 1. Which of the ordered pairs below satisfy the equation x y = 5? (-2, 3) (6, 1) (0, -5) (-3, -8) 2. Which of the ordered pairs below satisfy the equation 2x + 3y = 6? (0, 3) (6, -2) (3, 0) (-3, 4) 3. Write the equation x y = 5 in Slope-Intercept Form. 4. Write the equation 2x + 3y = 6 in Slope-Intercept Form.
12 Practice Problems 5. Draw an accurate graph of the linear equation 2x + 4y = 12. Slope-Intercept Form: Slope: Vertical Intercept: Horizontal Intercept: 6. Draw an accurate graph of the function 3x 2y = 10. Slope-Intercept Form: Slope: Vertical Intercept: Horizontal Intercept:
13 Practice Problems 7. Which of the ordered pairs below satisfy the linear inequality y > 3 x? (1,2) (0, 0) (5, 0) 8. Which of the ordered pairs below satisfy the linear inequality 3 y x 1? 5 (1,2) (0, 0) (5, 0) 9. Which of the ordered pairs below satisfy the linear inequality 4x y 3? (1,2) (0, 0) (5, 0) 10. Which of the ordered pairs below satisfy the linear inequality y 4? (1,2) (0, 0) (5, 0)
14 Practice Problems 11. Graph the solution sets of each of the following linear inequalities. a. y > 3 x b. y 3 5 x 1 c. 4x y < 3
15 Practice Problems d. x + y 5 e. y > x 2 f. y < 4
16 Practice Problems g. x 2 h. 5x 3y > 0 i. 2x < 6y
17 Practice Problems Applications 12. At a concession stand, three hot dogs and five sodas cost $ c. Let h represent the price of each hot dog, and s represent the price of each soda. Write a linear equation in general form to represent this situation. d. If hot dogs cost $3.25 each, how much is each soda? 13. The Science Museum charges $14 for adult admission and $11 for each child. The museum bill for a school field trip was $896. a. Write a linear equation in general form to represent this situation. Clearly indicate what each variable represents. b. Nine adults attended the field trip. How many children were there? 14. Bill begins a 50 mile bicycle ride. Unfortunately, his bicycle chain breaks, and he is forced to walk the rest of the way. Bill walks at a rate of 4 miles per hour, and rides his bike at a rate of 18 miles per hour. a. Let b represent the amount of time Bill spent bicycling before the chain broke, and w represent the amount of time Bill spent walking. Write a linear equation in general form to represent this situation. (Hint: Distance = rate time) b. Bill had been riding his bike for two hours when the chain broke. Use the equation in part a to determine the amount of time he spent walking.
18 Practice Problems Extension 15. *Refer to your course syllabus* a. The Final Exam for this class is worth % of your course grade. b. Let x represent the score you make on the Final Exam (as a percent), and y represent your grade in the class (as a percent) just prior to taking the Final Exam. Write a linear inequality in general form to represent this situation, assuming that you want your final course grade to be: A: At least 90% B: At least 80% C: At least 70% Hint: If your Final Exam is worth 30% of your course grade, then everything else would be worth 100% 30% = 70% of your course grade. c. Suppose you have a 77% in the class just before taking the final exam. What score do you need to make on the Final Exam to earn an A, B, or C in the class? Assume that your instructor does not round up!
19 Unit 11: Review 1. Draw an accurate graph of the linear equation 2x + 3y = 6. Determine the slope and intercepts of this linear equation and rewrite this equation in Slope-Intercept Form. Slope-Intercept Form: Slope: Vertical Intercept: Horizontal Intercept: 2. Draw and accurate graph of the solution set of linear inequality y 5 2x.
20 Review 3. Which of the ordered pairs below satisfy the linear inequality 2x 3y < 5? Circle all that apply. (2, 0) (1,-1) (0, 0) (2, 4) 4. Tickets to a movie cost $8.50 for adults and $6.00 for children. A total of $409 was collected in ticket sales for the 9:30AM show. a. Write an equation representing the total amount of money collected. b. If 37 children s tickets were purchased, how many adults were there?
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