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Solving Linear Inequalities: Classwork and Homework

This lesson focuses on solving linear and compound inequalities. Learn how to graph and interpret solutions using open and closed circles. Practice problems and classwork included.

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Solving Linear Inequalities: Classwork and Homework

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  1. Quiz 1.1-1.4 tomorrow Main Events: • Check: p. 1010 # 1-29 odd, [Extra Practice 1.1-1.6]

  2. Algebra 2Warm-up: 1/3/2020 Solve the equation. 3. Find the dimensions if the perimeter is 23 units. 4x+1 8-x 5x-2

  3. Answers • -2/9 • -16/5 • X=2; 6, 8, 9

  4. Section 1.6 • Date: ________ • Objective: Solve linear and compound inequalities.

  5. Lesson • Powerpoint 1.6 with practice • Special Cases with inequalities “and” “or” • Classwork: Puzzle D-27 • Show work on a clean sheet & staple • Will collect

  6. 1.6 Solving Linear Inequalities The solution of an inequality in one variable is any value that will make the inequality true. For example, the solution to x > –3 is any number greater than –3. We can use a number line to show all real numbers greater than (not including) –3. -3 To show that a point is not included, we used an open circle.

  7. Example: Graph x < 1 <means “less than or equal to” 1 We use a closed circle to show that a point is included. When graphing linear inequalities on a number line, > or < open circle> or < closed circle

  8. Is this statement true?2 < 3 What happens to the inequality if we add one to both sides? 3 < 4 What if we subtract one from both sides? 1 < 2 What if we multiply both sides by one? 2 < 3 What if we multiply both sides by –1? –2 > –3 Whenever an inequality is multiplied or divided by a negative, the inequality will reverse.

  9. Example 1: Solve – 4y + 8 > 20 –8 –8 –4y > 12 –4y12 –4 –4 When multiplying or dividing by a negative, the inequality switches direction. y < –3 –3

  10. To solve a compound linear inequality, you must isolate the variable between the two inequality signs. Example 2: 3 < 2x + 5 < 17 –5 –5 –5 –2 < 2x < 12 –2<2x<12 2 2 2 –1 < x ≤ 6 This solution is stating that x can be any real number between –1(not included) and 6 (included -1 6

  11. Solving Compound Inequalities 4 0 1 2 3 5 –1 Previously you studied four types of simple inequalities. In this lesson youwill study compound inequalities. A compound inequality consists of twoinequalities connected by and or or. Write an inequality that represents the set of numbers and graph the inequality. All real numbers that are greater than zero and less than or equal to 4. 0 < x£ 4 SOLUTION This inequality is also written as 0 < xandx£ 4.

  12. Solving Compound Inequalities 4 2 1 0 –1 3 –2 –1 0 1 2 3 4 5 Previously you studied four types of simple inequalities. In this lesson youwill study compound inequalities. A compound inequality consists of twoinequalities connected by and or or. Write an inequality that represents the set of numbers and graph the inequality. All real numbers that are greater than zero and less than or equal to 4. 0 < x£ 4 SOLUTION This inequality is also written as 0 < xandx£ 4. All real numbers that are less than –1or greater than 2. SOLUTION x < –1 orx > 2

  13. Compound Inequalities -two simple inequalities joined by “or” or “and” -“or” inequalities form a disjunction x3 or x0 -“and” inequalities form a conjunction x2 and x<5 (conjunctions may be written as one string ex: 2<x<5)

  14. Solving a Compound Inequality with And 7 6 3 2 1 0 4 5 The solution is all real numbers that are greater than or equal to 2 and less than or equal to 6. Solve –2 £ 3x – 8 £ 10. Graph the solution. SOLUTION Isolate the variable x between the two inequality symbols. –2 £ 3x – 8 £ 10 Write original inequality. 6 £ 3x£ 18 Add 8 to each expression. 2 £x£ 6 Divide each expression by 3.

  15. Solving a Compound Inequality with Or 3x + 1 < 4 or 2x – 5 > 7 1 0 2 3 4 6 5 7 –1 3x < 3 or 2x > 12 x < 1 or x > 6 The solution is all real numbers that are less than 1or greater than 6. Example 3: Solve 3x + 1 < 4 or 2x – 5 > 7. Graph the solution. SOLUTION

  16. Example 4: Reversing Both Inequality Symbols 0 3 –3 –2 –4 2 1 –5 –1 Multiply each expression by –1 and reverse both inequality symbols. Write original inequality. –2 < –2 – x < 1 Add 2 to each expression. 0 < –x < 3 0 > x > –3 –3 < x < 0. The solution is all real numbers that are greater than –3 and less than 0.

  17. Practice 1.6----Solve and graph:

  18. Classwork: D-27 Homework: p. 44 # 5-50(x5), 52 Closure: When do you use an open circle? When do you use a closed circle?

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