Discovering Algebra. Unit 4 Solving Inequalities & Systems of Inequalities Ch

Transcription

1 Discovering Algebra Unit 4 Solving Inequalities & Systems of Inequalities Ch

2 Unit 4: Linear Systems of Equations & Inequalities (Ch. 5) ACT Standards A 604. Solve systems of two linear equations Ch A 405. Match simple inequalities with their graphs on the number line (e.g., x ) Ch. 5.5 A 503. Solve first-degree inequalities when the method does not involve reversing the inequality sign Ch.5.5 A 602. Solve linear inequalities when the method involves reversing the inequality sign Ch A 603. Match linear inequalities with their graphs on the number line Ch

3 RECALL: Order of Operations Use the order of operations to simplify the following expressions ( 3) ( 8) 3. 5( 7 2) ( 5)

4 SOLUTIONS: Order of Operations Use the order of operations to simplify the following expressions ( 3) ( 8) ( 7 2) ( 5)

5 RECALL: Solve the following equations. 1. 3( x 7) x x ( x 8) 14 24

6 SOLUTIONS: Solve the following equations. 1. 3( x 7) x x = 14 3x 13 = 14 3x = 27 x = 9 3. x 4( x 8) x = -24 4x 18 = -24 4x = -6 x = -1.5

7 RECALL: Solve the following equations x x 12x x 9x 112 9x 5x 14 2x

8 1. SOLUTIONS: Solve the following equations. 3 x x 12x 90 15x = 90 x = x 9x x = -112 x = x 5x 14 2x 4x = x 2x = 14 x = 7

9 Lesson 5.5 Inequalities in One Variable Inequality: a statement that one quantity is less than or greater than another. Everyday Phrase Translation Inequality At least six glasses Below 40 Fewer than 20 calories At most $10 The number of glasses is greater than or equal to 6. The temperature is less than 40. The number of calories is less than 20. The price of the gift is less than or equal to $10. g 6 t < 40 c < 20 p 10 Between 35 and 120 The temp. is greater than 35 and less than 120. t > 35 and t < < t <

10 Lesson 5.5 Solving Linear Inequalities Inequalities are statements that include the following symbols: Is greater than Is less than Is greater than or equal to Is less than or equal to

11 Perform operation to BOTH sides. Does adding or subtracting affect the direction of the inequality sign?

12 SOLUTIONS: Does adding or subtracting affect the direction of the inequality sign? No, neither of these operations causes the direction of the inequality to switch. < < < < <

13 Perform operation to BOTH sides. Does multiplying or dividing affect the direction of the inequality sign?

14 SOLUTIONS: Does multiplying or dividing affect the direction of the inequality sign? YES, when multiplying or dividing by a negative number, the symbol MUST REVERSE its direction in order to be TRUE. < < < > < >

15 Solve & graph on a number line the following inequalities. 3 2x 3 1.) 2.) 7 2( x 3) 25 3.) 4.) x (5 3x)

16 SOLUTIONS: Solve & graph on a number line the following inequalities. 3 2x 3 1.) 2.) 2x x x -3-2x ( x 3) -2x x ) 10 2(5 3x) x 4.) x x 3.2x > x > x

17 Inequality Checks: 3 2x 3 3 2x 3

18 Solve & graph the following problems with a partner x ( x 8) 10 3

19 SOLUTIONS: Solve & graph the following problems with a partner x ( x 8) x x x x -30 x

20 Compound Inequalities (Intervals) Interval Inequality Graph Open 0 < x < Closed 0 x Neither 0 < x Neither 0 x < Verbal All real # s between 0 & 3 All real # s from 0 to 3 All real # s greater than 0 & less than or equal to 3 All real # s greater than 0 & less than 3 = closed circle (boundary is included in soln. set) = open circle (boundary is NOT included in soln. set)

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22 SOLUTIONS: x < -2 Neither -7 < s < 9 Open; shade in between -7 & 9-3 < x < 5 Open Closed; shade in between 4 & x 6 Closed -3 < u 3 Neither; shade in between -3 & 3

23 Compound Inequalities (continued) AND: the set of numbers that agree with BOTH inequality statements. (AND = OVERLAP AND = # s they have in common) OR: the set of numbers that agree with either inequality. (OR = Union) 7. 8.

24 7. Compound Inequalities (continued) SOLUTIONS: AND: the set of numbers that agree with BOTH inequality statements. (AND = OVERLAP AND = # s they have in common) OR: the set of numbers that agree with either inequality. (OR = Union) Open circle on 18; shade left. Open circle on 12; shade right. 8. Open circle on 12 & on 18; shade in between. Shade the entire number line. Closed circle on 64; Open circle on 72; shade in between. h 64 and h < 72

25 Solving Compound Inequalities Solve each compound inequality & graph. or -8 > -7k + 6 > -50

26 SOLUTIONS: Solving Compound Inequalities Solve each compound inequality & graph. or x < -6 or x > 8 (Open circle on -6; shade left. Open circle on 8; shade right) n 1 and x 7 or 1 x 7 closed circles on 1 & 7; shade in between) -10 < x < -3 (open circles on -10 & -3; shade in between) -8 > -7k + 6 > < k < 8 (open circles on 2 and 8; shade in between) x > 2 or x < -7 (open circle on 2; shade right. Open circle on -7; shade left)

27 Multi-Step Inequalities & Applications Ex.1.) You received $50 from your grandparents for your birthday. You make $7.50 each week for odd jobs around the neighborhood. Since your birthday, you have saved more than $120. How many weeks ago was your birthday? Define your variable(s). Write an inequality. Solve.

28 SOLUTIONS: Multi-Step Inequalities & Applications