Algebra 6.1 Solving Inequalities by Addition and Subtraction

Transcription

1 Algebra Chapter 6 Solving Linear Inequalities K: Students will know how to solve and graph inequalities and absolute value equations. U: Students will understand the similarities and differences between inequalities and linear equations. D: Students will solve inequalities using addition, subtraction, multiplication, and division. D: Students will solve compound inequalities. D: Students will solve and graph absolute value equations and inequalities. D: Students will graph inequalities and solve systems of inequalities. Algebra 6.1 Solving Inequalities by Addition and Subtraction Graph the following statements using a number line. x 15 x > 3 y 4 z < 6 p

2 Addition Property of Inequalities If any number is added to each side of a true inequality, the resulting inequality is also true. m - 19 original inequality add 19 to each side m 17 simplify The solution is the set {all numbers less than or equal to 17} A more concise way of writing a solution set is to use set-builder notation. {m m 17} Check the solution. To check, substitute 17, a number less than 17, and a number greater than 17. Graph it on a number line. Solve Inequalities by Addition 1) t ) b ) t - 16 > 4

3 Subtraction Property of Inequalities If any number is subtracted from each side of a true inequality, the resulting inequality is also true. x original inequality subtract 12 from each side x -8 simplify {x x -8} Check the solution. Graph it on a number line. Solve Inequalities by Subtraction 4) t ) b ) t + 16 > 4

4 Solve each inequality. Check your solution, and then graph it on a number line. 7) -7 + x 8) 9 < b + 4 9) 8r + 6 9r 10) 7p 5p - 2 Phrases that indicate an inequality < > less than greater than at most at least fewer than more than no more than no less than less than or equal to greater than or equal to

5 Define a variable, write an inequality, and solve each problem. Check your solution. 11) A number decreased by 8 is at most ) Twice a number is no less than -5 plus the number. 13) The sum of a number and 13 is at least ) 4 is no greater than a number plus 12.

6 Algebra 6.2 Solving Inequalities by Multiplication and Division Solve the following inequalities 1) t ) b ) 3b + 1 b Multiplication Property of Inequalities If each side of a true inequality is multiplied by the same positive number, the resulting inequality is also true. 1/4n 12 Original Equation x4 x4 Multiply by the reciprocal n 48 Simplify Division Property of Inequalities If each side of a true inequality is divided by the same positive number, the resulting inequality is also true. 1/4n 12 Original Equation x4 x4 Multiply by the reciprocal n 48 Simplify

7 Solve Each Inequality 1) b < ) a ) 6g ) 3n 3 Multiply and Divide by a negative number Divide each side by 3 Divide each side by 6 < 15 6 < ? 5? -5 What happened to the inequality when you divided by 3 and? 1) -5t 275 2) -1n < 12 4

8 Solve Each Inequality 1) 15 > t 2) -5g > 40 3) x < ) -8q < 136 Write an inequality and solve each problem. 1) Sports pennants are on sale for $2.50 each. Which inequality can be used to find how many sports pennants Amy can purchase for herself and her friends if she wants to spend no more than $15? A) 2.50p > 15 C) 2.50 < 15p B) 2.50p < 15 D) 2.50 > 15p 2) Bob walks a rate of 3 mile per hour. He knows that it is at least 9 miles 4 to Onyx Lake. How long will it take Bob to get there?

9 6 Solving Multi-Step Inequalities When solving a multi-step inequality you will need to add or subtract, then multiply or divide. Solve: 5b - 1 > -11 Original inequality Add 1 to both sides 5b > Divide both sides by 5 b > The solution set is {b b > } Ch. 6 Solving Linear Inequalities Solve: -7b + 19 < - 16 Original inequality Subtract 19 from each side -7b < Divide by -7 (Changes Inequality Symbol) b > 5 The solution set is {b b > 5} Solve y + 3 > n + 12 < n - 3 Solve using the distributive property. 3. 3(x + 4) > c + 3(2 - c) > c + 1

10 Solve Empty Sets -7(x + 4) + 11x > 8x - 2(2x + 1) -7x x > 8x - 4x - 2 4x - 28 > 4x - 2-4x - 4x - 28 > - 2 What happens when we work out an inequality and it results in a false statement? Is - 28 > a true statement? No, so we say that the Move solution me to find set the is answer empty, written as the symbol. (No number can fulfill this inequality) Solve 18-3(8c + 4) > -6(4c - 1) 18-24c -12 > - 24c c > - 24c c > - 24c c > c All Reals What happens when we work out an inequality and it results in a statement that is ALWAYS TRUE? c > c {c c is a real number} If solving an inequality Move results me to in find a the statement answer. that is always true, the solution set is the set of all real numbers.

11 Solve 1. 8(t + 2) - 3(t - 4) < 5(t - 7) w + 3 < (z + 1) + 11 > (z + 13) Write and Solve an Inequality 1. Three times a number minus eighteen is at least five times the number plus twenty-one. 2. Two more than half of a number is greater than twenty-seven.

12 6.4 Solving Compound Inequalities A compound inequality containing and is true only if both inequalities are true. Its graph is the intersection of the graphs of the two inequalities. For example: w > 40 and w < The Solution Set is {w 40 < w < 250 } Graph the solution set of x < 3 and x > Graph x < Graph x > Find the intersection Solution Set is {x < x < 3}

13 Graph the solution set of each compound inequality. 1. a > - 5 and a < 0 2. p < 6 and p > 2 Solve and Graph < x - 4 < 2 FIRST express - 5 < x - 4 < 2 using "and"

14 Solve and Graph 4. 6 < r + 7 < y A Compound Inequality containing or is true if one or more of the inequalities is true. Its graph is the union of the two inequalities. 6. x - 4 > 3 or x r + 1 < 7 or r - 3 > 6

15 Solve and graph h + 4 < 19 or 7h - 3 > k or 12-9k x 3 or -x x 3 and -x

16 12. x 9 and 2x < y + 3 < 3y -5 Write and Graph a Compound Inequality A pilot flying at 30,000 feet is told by the control tower that he should increase his altitude to at least 33,000 feet or decrease his altitude to no more than 26,000 feet to avoid turbulence. Write and graph a compound inequality that describes the optimum altitude (thousands)

17 Write and Graph a Compound Inequality A store is offering a $30 mail-in rebate on all color printers. Luisana is looking at different color printers that range in price from $175 to $260. How much can she expect to spend after the mail-in rebate? Compound Inequalities Summary AND = where graphs overlap x > and x < 4 x > 1 and x > 5 Solution: Solution: x < -1 and x > 3 x 2 and x 2 Solution: Solution:

18 OR = combination of both graphs (more) Ch. 6 Solving Linear Inequalities x < or x > 4 x > 1 or x > 5 Solution: Solution: x > -1 or x < 3 x 2 or x 2 Solution: Solution:

19 Algebra 6.5 Solving Open Sentences Involving Absolute Value Vocabulary distance from zero on a number line a function written as f(x) = x a function that can be written as 2 or more expressions Find the absolute value 1) -5 2) 9 3) ) -9-12

20 Solve and graph on a number line: Solve x-12 = 24 Solve b+9 = 2

21 Solve and graph on a number line: Solve x - 5 = 8 Solve h + 9 = 13 Solve r - 12 = 1 Solve y + 3 = -6 Now let's work backwards! How do we go from the graph above to the absolute value equation below? r - 12 = 1

22 ...??? Write the equation as: x - (midpoint) = (distance from point to midpoint) So... x - 1 = 5 Write an open sentence involving absolute value equation for each graph

23 Graph f(x) = x + 3 (review) x x + 3 But, what happens when there are absolute value symbols? x x + 3 Graph f(x) = x + 3 Since x + 3 cannot be negative, f(x) cannot be negative and therefore the graph of x + 3 cannot go below the x-axis. This means that there is a minimum point of the line where it "bounces" off the x-axis (the x-intercept). To find the minimum point, substitute zero for f(x) and make it equal to the equation of the line inside the absolute value symbols and solve for x. 0 = x + 3 x = This value is your minimum point. Use chart to plug in values less than and greater than the minimum point. x x + 3

24 What is the domain and range? Graph f(x) = 3x - 9. What is the domain and range?

25 Online Graphing Calculator Graph g(x) = x What is the domain and range? Graph f(x) = x What is the domain and range?

26 Algebra 6.6 Solving Inequalities Involving Absolute Value Review of Solving Absolute Equations Write an equation involving Absolute Value Ch. 6 Solving Linear Inequalities Find the solution set and graph t + 5 < 9 t < 4 -(t + 5) < 9 t + 5 > -9 t >

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29 6-7 Graphing Inequalities in Two Variables Solve the inequality and graph the solution set: a) 3 < 2x - 3 < 15 b) 3n or n -12 c) Graph y = Graph y > Half Plane - The graph of ordered pairs that fill a region on the coordinate plane. Boundary - An equation defines the boundary or edge for each half-plane. **If the inequality symbol is < or >, draw the boundary as a dashed line. **If the inequality symbol is < or >, draw the boundary as a solid line.

30 2. Graph y - 2x < - 4 Solve for y Graph as if it was an equation Shade the half-plane that is the solution (test points) Graph x < Graph y > 1x

31 5. Graph 2y - 4x > Graph 4-2x < Write and Graph an Inequality Rose sells radio advertising in 30-second and 60-second time slots. During every hour, there are up to 15 minutes available for commercials. How many commercial slots can she sell for one hour of broadcasting?

32 Write and Graph an Inequality Lee Cooper writes and edits short articles for a local newspaper. It takes him about an hour to write an article and about a half-hour to edit an article. If Lee works up to 8 hours a day, how many articles can he write in one day?

33 6-8 Graphing Systems of Inequalities Graph just as if it was an equation plot the y-intercept use slope to get multiple points System of Inequalities is a set two or more inequalities with the same variables. To solve you find the ordered pairs that satisfy all inequalities The solution set is represented by the intersection, or overlap of the graphs Review of 6.7: Graph Graph What does the shaded area of an inequality mean? What does the area where the shading overlaps mean? Solve by Graphing

34 Solve by Graphing Ch. 6 Solving Linear Inequalities Solve by Graphing y < 2x + 2 y > - x - 3 y > - 3x + 1 y < - 3x - 2

35 2x + y > 2 2x + y < 4 Use a Systems of Inequalities to solve The middle 50% of first-year students attending the University of Massachusetts at Amherst scored between 520 and 630, inclusive, on the math portion of the SAT. They scored between 510 and 620, inclusive, on the critical reading portion of the test. Graph the scores that a student would need to be in the middle 50% of first year students.

36 Use a Systems of Inequalities to solve 8. The LDL or "bad" cholesterol of a teenager should be less than 110. The HDL "good" cholesterol of a teenager should be between 35 and 59. Make a graph showing appropriate levels of cholesterol for a teenager.

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