Presentation on theme: "Lesson 1-5 Solving Inequalities September 24 - 25 Objective: The students will be able to solve, algebraically and graphically, inequalities."— Presentation transcript:
Lesson 1-5 Solving Inequalities September 24 - 25 Objective: The students will be able to solve, algebraically and graphically, inequalities
When an inequality is greater than or less than, and NOT equal to, use an open dot on the graph. When an inequality includes an “equal to”, use a closed dot on the graph.
Notice than when the variable is by itself on the left side of the inequality, the shaded part of the graph points the same direction as the greater than or less than symbol in the inequality
What inequality represents the sentence, “5 fewer than a number is at least 12.”? 1)What does “fewer than” mean? Subtraction 2)What does “is at least 12” mean? Greater than or equal to x - 5 ≥ 12
What inequality represents the sentence, “The quotient of a number and 3 is no more than 15.”? 1)What does “quotient” mean? Division 2)What does “is no more than” mean? Less than or equal to The quotient of a number and 3 means a number divided by 3, with an answer less than or equal to 15. x/3 ≤ 15
The solutions of an inequality are the numbers that make it true. The properties you use for solving inequalities are similar to the properties you use for solving equations. However, when you multiply or divide each side of an inequality by a negative number, you must reverse the inequality symbol. For a detailed proof of why this is true, see page 34 of your textbook.
This chart was captured straight from your textbook. It is found on page 34.
What is the solution of -3(2x-5) + 1 ≥ 4? Graph the solution. What should we do first? Distribute the -3 -6x + 15 +1 ≥ 4 What’s next? Combine like terms (simplify) -6x + 16 ≥ 4 Now what? Isolate x Subtract 16 from both sides -6x ≥ -12
Can you see the finish line? Divide both sides by -6 to find x -6x ≥ -12 x ≤ 2 Remember to switch the inequality symbol because we divided by a negative number. What does the graph look like?
How do we check the solution of -3(2x-5) + 1 ≥ 4? Plug in a number that is less than or equal to 2 and make sure that we get a true statement. What number should we use? Why not zero? -3(2(0)-5) + 1 ≥ 4 -3(-5) + 1 ≥ 4 15 + 1 ≥ 4 16 ≥ 4 True statement, answer checks.
A digital music service offers two subscription plans. The first has a $9 membership fee and charges $1 per download. The second has a $25 membership fee and charges $.50 per download. How many songs must you download for the second plan to cost less than the first plan?
Let’s assign a variable. Let’s let s represent the number of songs. Now we need to write two expressions, one for each plan. What algebraic expression states the terms of plan 1? 9 + s Which algebraic expression represents the second plan? 25 +.5s
What is the relationship between the two plans? We want the second plan to be less than the first plan, or Plan A to be greater than Plan B. Our mathematical problem will look like. 9 + s > 25 +.5s Solve this now, at your desk. What did you get? The correct answer is more than 32 songs. s > 16 +.5s (take 9 from each side).5s > 16 (take.5s from each side) s > 32 (divide both sides by.5) If you will use the service more than 32 times, go with plan B.
Just like with equalities, some inequalities are always true, some are never true, and the rest are sometimes true. -2(3x + 1) > -6x + 7 -6x -2 > -6x + 7 -2 > 7 Since this is a false statement, this inequality is never true
Just like with equalities, some inequalities are always true, some are never true, and the rest are sometimes true. 5(2x - 3) - 7x ≤ 3x + 8 10x -15 - 7x ≤ 3x + 8 3x - 15 ≤ 3x + 8 -15 ≤ 8 Since this is a true statement, this inequality will always be true.
Sometimes, we need to solve more than one inequality. There are AND inequalities, and OR inequalities. 7 < 2x +1 AND 3x ≤ 18 6 < 2xx ≤ 6 3 < x x > 3 AND x ≤ 6 That means that all numbers between 3 and 6, including 6, are solutions to both inequalities.
Sometimes, we need to solve more than one inequality. There are AND inequalities, and OR inequalities. 7 + k ≥ 6 OR 8 + k < 3 k ≥ -1 k < -5 OR means that a solution makes either inequality true, so the solutions include all numbers EXCEPT those between -1 and -5, including -1.
Classwork: Page 38 #10-42 Even Only Homework: Pages 39-40 #44-64 Even MAKE SURE THAT YOU HAVE FINISHED WORKBOOK PAGES 7-8, 11-12, and 15-16. Tutoring this week will be Tuesday and Thursday from 2:30 to 4:00