Math Practice 3-1 Homework Inequalities And Their Graph

This is Timmy. Timmy is the best man at his friend's wedding, but he is running super late. The road he's driving on has a speed limit of 55 miles per hour, which means he should not drive more than 55 miles per hour. This can be represented by an inequality.

Graphing Inequalities: x < y

Let's take a closer look, how to graph this inequality on a number line. The inequality representing speeds less than 55 mph is x < 55. To graph this on the number line, we draw an open circle at 55 mph and an arrow to the left, which shows all speeds less than 55mph.

However, Timmy could also drive at exactly 55 mph, which is the posted speed. The inequality representing speeds 55mph and under is: x is less than or equal to 55mph. We can show this on the number line by using a closed circle at 55 mph instead of an open one, along with the arrow to the left that represents all speeds less than 55mph. Our graph represents the speeds Timmy could drive.

And remember: we use a closed circle because 55mph is included in the inequality, since Timmy could drive at exactly 55mph.

Graphing Inequalities: x > y

Timmy really has to hurry because he's got the wedding cake! Could he go faster than 55mph? Not without breaking the law and risking getting a speeding ticket for driving too fast. We can represent speeds that are faster than 55mph with the inequality x > 55.

The graph of this inequality has an open circle at 55mph and arrow to the right. This graph shows the speeds that would earn Timmy a speeding ticket, since all of them are faster than the posted speed limit.

Graphing Inequalities: x > −y

Now that we understand the main idea of inequality graphs, let's look at another example. This inequality is x > −20.

To graph this, we must include all the numbers greater than, but not equal to, −20. Therefore, we draw an open circle at −20 and an arrow to the right.

If the inequality were x is less than or equal to −20, then we'd need to change two things on the graph.

First, −20 is included in the inequality, so we use a closed circle. Second, this inequality consists of numbers less than or equal to −20, so we use an arrow to the left.

Oh no! Even though Timmy obeyed the speed limit, he was in such a hurry that he didn't notice a bump in the road. Now the cake is ruined!

Presentation on theme: "~ Chapter 3 ~ Algebra I Algebra I Solving Inequalities"— Presentation transcript:

1 ~ Chapter 3 ~ Algebra I Algebra I Solving Inequalities
Lesson Inequalities & Their GraphLesson Solving Inequalities Using Addition & SubtractionLesson Solving Inequalities Using Mult. & Div.Lesson Solving Multi-Step InequalitiesLesson Compound InequalitiesLesson Absolute Value Equations & InequalitiesChapter Review

2 Inequalities & Their Graph
Lesson 3-1Inequalities & Their GraphCumulative Review Chap. 1-2

3 Inequalities & Their Graph
Lesson 3-1Inequalities & Their GraphExtra Practice - Chap 2

4 Inequalities & Their Graph
Lesson 3-1Extra Practice - Chap 2

5 Inequalities & Their Graph
Lesson 3-1Extra Practice - Chap 2

6 Inequalities & Their Graph
Lesson 3-1Inequalities & Their GraphNotesInequality – mathematical sentence that contains >, <, ≥, ≤, ≠.Solution of an inequality – any number that makes the inequality true.Example ~ y > 5 Would 6 make the inequality true? What about 7, 4, 9, 22, -5?Graphing solutions of InequalitiesGraph y < 3Graph x > -1Graph a ≤ -2Graph -6 ≤ g

7 Inequalities & Their Graph
Lesson 3-1Inequalities & Their GraphNotesGraph ½ ≥ cWriting Inequalities to Describe Graphs…Define a variable and write an inequalityA bus can seat at most 48 studentsYou must be at least 16 years old to obtain a driver’s licenseHomework – Practice 3-1

8 Solving Inequalities Using Addition & Subtraction
Lesson 3-2Practice 3-1

9 Solving Inequalities Using Addition & Subtraction
Lesson 3-2Practice 3-1

10 Solving Inequalities Using Addition & Subtraction
Lesson 3-2NotesEquivalent Inequalities – Inequalities with the same solution. Ex ~ x > 2 and x – 5 > -3Steps for solving an inequality(1) Divide the equation at the inequality sign into two equal sides(2) Underline the variable.(3) Identify the number on the same side as the variable.(4) Identify the operation (addition or subtraction) and perform the opposite (inverse operation) to both sides of the equation.(5) Simplify and solve the inequality(6) Check your answer.x + 7 > 22 m – 5 < -61x > m < -56Graph the solutions

11 Solving Inequalities Using Addition & Subtraction
Lesson 3-2Notes-3.3 ≤ x t - 5 ≥ 11-10.8 ≤ x t ≥ 16Graph the solutionsHomework – Practice 3-2 # 1-20

12 Solving Inequalities Using Addition & Subtraction
Lesson 3-2Practice 3-2

13 Solving Inequalities Using Addition & Subtraction
Lesson 3-2Practice 3-2

14 Solving Inequalities Using Addition & Subtraction
Lesson 3-2NotesWrite & Solve the Inequality that models each situation.Your baseball team has a goal to collect at least 160 blankets for a shelter. Team members brought 42 blankets on Monday and 65 blankets on Wednesday. How many blankets must the team donate on Friday to make or exceed their goal?Your brother has $2000 saved for a vacation. His airplane ticket is $637. How much can he spend for everything else?Homework – Practice 3-2 #21-42

15 Solving Inequalities Using Mult. & Div.
Lesson 3-3Practice 3-2

16 Solving Inequalities Using Mult. & Div.
Lesson 3-3Practice 3-2

17 Solving Inequalities Using Mult. & Div.
Lesson 3-3NotesSteps for solving an inequality(1) Divide the equation at the inequality sign into two equal sides(2) Identify the variable.(3) Identify the number on the same side as the variable.(4) Identify the operation (multiplication or division) and perform the opposite (inverse operation) to both sides of the inequality.(5) Simplify and solve the inequality(If you multiply or divide each side of an inequality by a negative number, you reverse the inequality symbol. )(6) Check your answer.11x > m/5 < -6÷ ÷ x5 x5x > m < -30Graph the solutions

18 Solving Inequalities Using Mult. & Div.
Lesson 3-3Notes-¾ b ≤ ≤ -0.9 px(-4/3) x(-4/3) ÷(-0.9) ÷(-0.9)b ≥ ≥ p (another way to solve?)Graph the solutionWrite & Solve the Inequality that models each situation.Students in the school band are selling calendars. They earn $0.40 on each calendar they sell. Their goal is to earn more than $327. Write and solve an inequality to find the fewest number of calendars they can sell and still reach their goal.Suppose you earn $8.15 per hour working part time at the dry cleaner. Write and solve an inequality to find how many full hours you must work to earn at least $100.Homework ~ Practice 3-3 odd

19 Solving Multi-Step Inequalities
Lesson 3-4Practice 3-3

20 Solving Multi-Step Inequalities
Lesson 3-4Practice 3-3

21 Solving Multi-Step Inequalities
Lesson 3-4Practice 3-3

22 Solving Multi-Step Inequalities
Lesson 3-4Notes 3-4Solving inequalities with variables on one side-3x -4 ≤ < 7 – 2t-3x ≤ < -2t÷ (-3) ÷(-3)(reverse) ÷(-2) ÷(-2) (reverse)x ≥ > t or t < 1Now you solve some…-8 < 5n – – 5k ≤ 23 < n k ≥ 2Distributive Property & Inequalities4p + 2(p + 7) < ≤ 5 – 2(4m + 7)4p + 2p + 14 < ≤ 5 – 8 m – 146p + 14 < ≤ -8m - 9Then… Solve like other multi step inequality

23 Solving Multi-Step Inequalities
NotesLesson 3-4Solving inequalities with variables on both sides6z – 15 < 4z (4 – m) ≥ 4(2m + 1)z < m ≤ 8/11Your turn…3b + 12 > 27 – 2b -6(x – 4) ≥ 7(2x – 3)b > ≥ x or 2 ¼ ≥ xWrite & Solve an inequalityOne half the difference of t and six is less than or equal to four½(t – 6) ≤ 4The perimeter of an isosceles triangle is at most 27 cm. One side is 8 cm long. Find the possible length of the two congruent sides.Homework Practice 3-4 odd

24 Compound Inequalities
Lesson 3-5Practice 3-4

25 Compound Inequalities
Lesson 3-5Practice 3-4

26 Compound Inequalities
Lesson 3-5NotesCompound Inequalities – Two inequalities joined by the word and or or.For example x > -6 and x < 8… How could we write this?-6 < x < 8 Graph?The solution for “and” joined inequalities is the overlap of the two graphs… i.e. where both graphs show the same solutions.Write and graph the compound inequalitityAll real numbers greater than -2 but less than 9-2 < x < 9The books were priced between $3.50 and $6.00, inclusive.3.50 ≤ c ≤ 6.00Solving a compound inequality containing and…Solve each inequality… then simplify-6 ≤ 3x < solve ≤ 3x and x < 15-2 ≤ x x < 5

27 Compound Inequalities
Lesson 3-5Notes7 < -3n + 1 ≤ Solve & Graph…7 < -3n and n + 1 ≤ Solution: -2 > n and n ≥ -4 or -4 ≤ n < -2Writing compound Inequalities with orDiscounted fares are available to children 12 and under or to adults at least 60 years of age.a ≤ 12 or a ≥ 60 Graph the solution…What else do we know?Write an inequality that represents all real numbers that are at most -5 or at least 3. Graph your solution.Solving a compound inequality containing or-2x + 7 > 3 or 3x – 4 ≥ 5 Graph the solutionx < or x ≥ 3Homework Practice 3-5 odd

28 Absolute Value Equations & Inequalities
Lesson 3-6Practice 3-5

29 Absolute Value Equations & Inequalities
Lesson 3-6Practice 3-5

30 Absolute Value Equations & Inequalities
Lesson 3-6NotesAbsolute Value – distance a number is away from 0.Solving an absolute value equation|x| + 5 = 11 |t| - 2 = -1|x| = |t| = 1x = 6 & x = t = 1 & t = -13|n| = = 3|w| - 2|n| = = |w|n = 5 & n = w = 2 & w = -2More absolute value equationsSometimes an absolute value equation has the expression inside the absolute value symbols.Solving Absolute Value Equations ~ To solve an equation in the form |A| = b, where A represents a variable expression and b > 0, solve A = b and A = -b.

31 Absolute Value Equations & Inequalities
Lesson 3-6Notes|c - 2| = 6 this means… c - 2 = 6 or c - 2 = Why?Solve … c = c = -4Your turn…-5.5 = |r + 2| |y - 3| = 9

32 Absolute Value Equations & Inequalities
Lesson 3-6NotesSolving Absolute Value Inequalities|n - 2| < 5 (represents all numbers whose distance from 2 is less than 5 units)So… -5 < n - 2 < 5 Graph the solution|n - 2| > 5 (represents all numbers whose distance from 2 is more than 5 units)So… n – 2 < -5 or n-2 > Graph the solutionHere are the rules…Rule 1 ~ To solve an inequality in the form |A| < b, where A is a variable expression and b > 0, solve –b < A < b.Rule 2 ~ To solve an inequality in form |A| > b, where A is a variable expression and b > 0, solve A < -b or A > b.Similar rules are true for |A| ≤ b or |A| ≥ b.

33 Absolute Value Equations & Inequalities
Lesson 3-6Practice 3-6

34 Absolute Value Equations & Inequalities
Lesson 3-6NotesSolve & graph the solutions… |v - 3| ≥ 4 Rule 1 or 2?v – 3 ≤ or v – 3 ≥ 4v ≤ v ≥ 7Solve & graph the solutions… |w + 2| < 5 Rule 1 or 2?-5 < w + 2 < < w and w + 2 < 5-7 < w and w < 3 … < w < graph…Write an absolute value inequality and solveAll numbers less than 3 units from 0|n| < 3The ideal diameter of a gear for a certain type of clock is mm. An actual diameter can vary by 0.06 mm. Find the range of acceptable diameters.|d – 12.24| ≤ Rule 1 or 2?-0.06 ≤ d – ≤ 0.06Homework ~ Practice 3-6#1-28 even & 29-36

35 Absolute Value Equations & Inequalities
Lesson 3-6Practice 3-6

36 Absolute Value Equations & Inequalities
Lesson 3-6Practice 3-6

37 ~ Chapter 3 ~Algebra IAlgebra IChapter Review

38 ~ Chapter 3 ~Algebra IAlgebra IChapter Review

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