Chapter Two B: Linear Expressions, Equations, and Inequalities Index: A: Intro to Inequalities (U2L8) Page 1 B: Solving Linear Inequalities (U2L9) Page 7 C: Compound Inequalities (And) (U2L10/11) Page 13 D: Compound Inequalities (Or) (U2L10/11) Page 19 E: Interval Notation (U2L12) Page 24 F: Modeling Inequalities (U2L13) Page 30 1
Name: Date: Period: Algebra I Intro to Inequalities 2B A So far we have concentrated on solving equations. Remember, all solving an equation consisted of was finding values of the variable that made the two expressions equal (in other words made the equation true). We can also judge the truth value of a statement that is in the form of an inequality. Exercise 1: For each inequality, state whether it is true or false. (a) 7 > 3 (b) 0 < 10 (c) 9 > 12 (d) 4 4 (e) 2 > -7 (f) -3-5 (g) -4 > 0 (h) 3 There are a lot of ways to formally define how to determine if one number if greater than another. We will use a graphical definition. Exercise 2: Give the truth value for each of the following statements. Draw a number line to support your answer. (a) 3 > -4 (b) -5 > -3 (c) 0 > -6 So since we can test the equality of numbers now, we can also test the equality of expressions for values of variables. This is identical to checking the truth value of an equation. Exercise 3: Given the inequality, determine if it is true or false for the following values of x. (a) (b) (c) 2
Notice that unlike equations, inequalities tend to have many values that make them true. We will eventually discuss that certain inequalities even have an infinite number of values for their variables that make them true. Exercise 4: Determine if the following inequality is true or false for the given value of x. for Exercise 5: Determine if the following inequality is true or false for the given value of x. for Exercise 6: Determine if the following inequality is true or false for the given value of x. for 3
Name: Date: Period: Algebra I Intro to Inequalities 2B A Homework 1) For each inequality, state whether it is true or false. 2) For each of the following inequalities, determine if it is true or false at the given value of the replacement variable. (a) for (b) for (c) for (d) for (e) for (f) for 4
3) A pressure gage for a boiler allows the boiler to run as long as psi, where x is the pressure reading at the sensor. If the pressure gets too high the machine will shut down to precent any injuries but it will also cost the company money. Test the following values to see what pressures will be safe for the machine to run at. (b) If the machine cannot run unless it has a pressure above 35 pounds per cubic inch, test to see if a reading of 5 would keep the machine functional. 4) Write the appropriate inequality sign (< or >) in the box that will make each of the following true at the given point. (Hint: Substitute into each expression and then pick if it is < or >.) 5
Review Section: 5) If, then what is the value of x? 6) What is the value of in the equation? 7) The length of a rectangle is represented by, and the width is represented by. Express the perimeter of the rectangle as a trinomial. Express the area od the rectangle as a trinomial. (Hint: It may help to draw/label a diagram.) 6
Name: Date: Period: Algebra I Solving Linear Inequalities 2B B Just like we can solve linear equations by using properties of expressions (commutative, distributive, and associative) and equations (addition and multiplication properties), we can do the same for inequalities. Before we solve, we have to look at some rules first. Let's look at some tests to discover these rules: Consider the true inequality: 4 < 8 If we... (a) Add 3 to both sides: (b) Subtract 4 on both sides: 4 < 8 4 < 8 4 +3 < 8 + 3 4-4 < 8-4 7 < 11 0 < 4 IT IS STILL TRUE! IT IS STILL TRUE! (c) Multiply by 2 to both sides: (d) Divide by 2 on both sides: 4 < 8 4 < 8 2(4) < 2(8) (4/2) < (8/2) 8 < 16 2 < 4 IT IS STILL TRUE! IT IS STILL TRUE! Based on the examples before, you might conclude that the truth values of inequalities have the same truth values for equalities (equations). However, there is one HUGE difference between linear equalities and linear equations. Let's find it! Returning to the true inequality 4<8 If we... (a) Multiply by -2 on both sides: (b) Divide by -4 on both sides: 4 < 8 4 < 8-2(4) < -2(8) (4/-4) < (8/-4) -8 < -16-1 < -2 IT IS NOW FALSE! IT IS NOW FALSE! Or in other terms, if we multiply or divide an inequality equation by a negative number, we must flip the sign in order for it to hold true! EXAMPLE 7
Let's try and practice solving linear equations: Exercise 1: Solve the following inequalities. (a) (b) (c) (d) Now let s graph inequalities! Here are some basic rules before we start: Exercise 2: Given the linear inequality (a) Solve the inequality answer the following: (b) Graph the inequality on the number line below (c) Name 5 numbers that are in the Solution Set (make the solution true) 8
Exercise 3: Given the linear inequality (a) Solve the inequality answer the following: (b) Graph the inequality on the number line below (c) Name 5 numbers that are in the Solution Set (make the solution true) Exercise 4: Given the linear inequality (a) Solve the inequality answer the following: (b) Graph the inequality on the number line below (c) Name 5 numbers that are in the Solution Set (make the solution true) 9
Name: Date: Period: Algebra I Solving Linear Inequalities 2B BHomework 1) Which number is in the solution set of (1) (2) (3) (4) 2) Which graph represents the inequality (1) (2) (3) (4) 3) Solve the inequality using the properties of inequalities and graph the final solution on the number line provided. (a) (b) (c) 10
(d) (e) 4) Two siblings, Edwin and Rhea, are both going skiing but choose different payment plans. Edwin s plan charges $45 for rentals and $5.25 per lift up the mountain. Rhea s plan was a bundle where her entre day cost $108. (a) Set up an inequality that models the number of trips,, up the mountain for which Edwin will pay more than Rhea. Solve the inequality. (b) What is the greatest amount of trips that Edwin can take up the mountain and still pay less than Rhea? Explain how you arrived at your answer. 11
5) Given a, b, c, and d are all positive, solve the following inequality for x. 6) Given determine the largest integer values of when. 7) Solve the inequality below to determine and state the smallest possible value of x in the solution set. Review Section: 8) The difference between twice a number a number that is 5 more than it is 3. Which of the following equations could be used to find the value of the number,? (1) (2) (3) (4) 9) Think about the equation (a) Solve this equation by reversing what has (b) Solve this equation by first distributing the been done to x. multiplication by 4. 12
Name: Date: Period: Algebra I Compound Inequalities (And) 2B C Graphing Compound Inequalities: Compound inequalities are problems that have more than one inequality that have to be graphed together. There are two different types that we need to understand: And Problems Or Problems We will do And problems today! AND: Both must be true (for the number to be a part of the solution set, it must satisfy both parts of the compound inequality). Graph both and keep the INTERSECTION. Exercise 1: On the number lines below, shade in all the values of x that solve the compound inequality. In other words, shade in the compound inequalities solution set. If you need a good place to start, try listing some x values that make the compound inequalities true. (a) and List some values: (b) List some values: (c) and List some values: 13
Exercise 2: Solve and graph the following compound inequality. Exercise 3: Solve and graph the following compound inequality. Exercise 4: Consider the compound inequality given by: and (a) Determine whether each of the following values of x falls in the solution set to this compound inequality. Show the work that leads to each answer. 14
(b) Solve the compound inequality and graph its solution on the number line shown below. and 15
Name: Date: Period: Algebra I Compound Inequalities (And) 2B C Homework 1) Determine if each of the following values of is in the solution set to the compound inequalities given below? Justify each of your choices by showing your calculations. (a) for: and (b) for: and 2) Solve and graph the following compound inequalities. (a) (b) 16
(c) (d) (e) 17
3) 4) Review Section: 5) What is the solution of? (1) (2) (3) (4) 6) 7) The expression is equivalent to? 18
Name: Date: Period: Algebra I Compound Inequalities (Or) 2B D Graphing Compound Inequalities: Compound inequalities are problems that have more than one inequality that have to be graphed together. There are two different types that we need to understand: And Problems Or Problems We will do Or problems today! OR: One must be true. (If the number satisfies either part of the compound inequality, or both parts, it is part of the solution set). Graph both and leave it alone. Exercise 1: On the number lines below, shade in all the values of x that solve the compound inequality. In other words, shade in the compound inequalities solution set. If you need a good place to start, try listing some x values that make the compound inequalities true. (a) or List some values: (b) or List some values: Exercise 2: Solve and graph the following compound inequality. 19
Exercise 3: Solve and graph the following compound inequality. Exercise 4: Consider the compound inequality shown below: or (a) Determine whether each of the following values of x falls in the solution set to this compound inequality. Show the work that leads to each answer. (b) Solve this compound inequality and graph the solution on the number line. What can you say about the solution set of this inequality? or 20
Name: Date: Period: Algebra I Compound Inequalities (Or) 2B D Homework 1) Determine if each of the following values of is in the solution set to the compound inequalities given below? Justify each of your choices by showing your calculations. (a) for: or (b) for: or 2) Write a compound inequality for each of the graphs. (a) (b) (c) 3) Solve and graph the following compound inequalities. (a) 21
(b) (c) (d) 22
(e) Review Section: 4) Which of the following is not an equation? (1) (2) (3) (4) 5) Find three consecutive odd integers such that the sum of the smaller two is three times the largest increased by seven. 23
Name: Date: Period: Algebra I Interval Notation 2B E We will often want to talk about continuous segments of the real number line. We ve already done work with this in the last lesson using what is known as inequality or set builder notation. Today we will see a very simple way of showing these segments. By Interval Notation: An interval is a connected subset of numbers. Interval Notation is an alternative to expressing your answer as an inequality. Exercise 1: For each of the following, write the equivalent using interval notation. (a) (b) Equivalent Interval Notation: Equivalent Interval Notation: (c) (d) Equivalent Interval Notation: Equivalent Interval Notation: (e) (f) Equivalent Interval Notation: Equivalent Interval Notation: 24
One of the great advantages of interval notation is that we essentially need to know a starting value, an ending value, and then whether they are included or not. Exercise 2: Pick the answer choice that best answers each question. 1) All numbers greater than or equal to positive ten. (a) (b) (c) (d) 2) All numbers from negative three up to an including positive seven. (a) (b) (c) (d) 3) Which interval notation represents the set of all numbers from 2 through 7 inclusive? (a) (b) (c) (d) 4) What is the difference between the number line graphs of these two intervals: (2,8] and [2,8]? (a) The first graph has an open circle at 2 while the second graph has a closed circle at 2. (b) The first graph has a closed circle at 2 while the second graph has an open circle at 2. (c) There is no difference between the graphs. 5) Which of the following represents the equivalent interval to? (a) (b) (c) (d) Exercise 3: Solve the inequality below for all values of x. Graph the solution on the number line given and state the solution set using interval notation. Exercise 4: Two inequalities have solution sets given in the interval notation below. Inequality #1: [-3,2) Inequality #2: (0,4) Interval Notation: 25
Exercise 5: At a hydroelectric plant, Pump #1 is on for all times on the interval [0,8) and Pump #2 is on for all times in the interval [4,18). Which of the following represents all times, t, when both pumps are on? (Hint: Sketch on number line). (1) (3) (2) (4) 26
Name: Date: Period: Algebra I Interval Notation 2B E Homework 1) Which of the following represents the set {x: x -3}? (a) (-3, ) (b) (-, -3) (c) [-3, ) (d) (-, -3] 2) Which of the following represents the solution to the inequality? (a) (, -2) (b) (-, -2] (c) (-, 2) (d) (-, 2] 3) Which of the following represents all the real numbers from 2 to 9, but not 2? (a) (2,9) (b) [2,9] (c) [2,9) (d) (2,9] 4) Write sets using interval notation for the sections of the number lines shown graphed below. (a) (b) Equivalent Interval Notation: Equivalent Interval Notation: 5) For each of the following, graph the portion of the number line described by the inequality and then write the equivalent using interval notation. (a) (b) Equivalent Interval Notation: Equivalent Interval Notation: (c) (d) 27
Equivalent Interval Notation: Equivalent Interval Notation: (e) or (f) and Equivalent Interval Notation: Equivalent Interval Notation: 6) Aiden wrote the interval (-5,4] and claimed it was equivalent to the graph below. Explain what he did wrong and correct his mistake. Review Section: 7) 28
8) Solve algebraically: If is a number in the interval [4,8], state all integers that satisfy the given inequality. Explain how you determined these values. 29
Name: Date: Period: Algebra I Modeling Inequalities 2B F Just as we can solve many real-world problems involving linear equations, there are plenty of situations when an inequality is called for instead. In this lesson, we will practice setting up and solving inequalities based on realworld scenarios. Exercise 1: A school is taking a field trip with 195 students and 10 adults. Each bus can hold at most 40 people. We need to determine the smallest number of busses needed for the trip. (a) Using a guess-and-check method, determine the minimum number of busses needed. Show evidence to your thinking. (b) Let n be the number of busses taken on the trip. Write and solve an inequality that models this problem based on n. It is important that you are able to deal with the phrases at least and at most. Let's try some translating, but first let's look at the definitions: Exercise 2: Translate each of the following phrases into an inequality. Do not solve. (a) When three times a number is increased by 12, the result is at least 32. (b) The sum of two consecutive even integers, and, is at most 8. 30
Exercise 3: Find all numbers for which five less than half the numbers is at least seven. Set up an inequality, carefully define expressions, and solve the inequality. Exercise 4: Find all numbers such that twice the sum of a number and eight is at most four. Solve this problem by setting up and solving an inequality. Now let's try to model a real-world scenario with an inequality! Exercise 5: A stadium is steadily filling up with people. It holds at most 2,500 people. Of the 2,500 seats, 350 are reserved for special guests. When the doors open, people fill the seats at a rate of 10 seats per minute. (a) If represents the number of minutes that have gone by, fill out the following chart for how many seats have either been taken or reserved. 31
(b) Write an expression that calculates the number of seats filled and reserved in terms of minutes, passed., that have (c) Write an inequality that shows times, in minutes, before the stadium is over-filled. Solve the inequality. (d) At the rate that people are entering, will any more people be able to find a seat after four hours? Justify your yes/no answer. (e) To cover the cost of the stadium, labor, and other overhead costs, stadium organizers must raise at least $39,000 from ticket sales. If they sell tickets at $25 each, will they have covered the cost if 1,250 tickets are sold? (f) Let represent the number of tickets sold. Write and solve an inequality that represents all values of that guarantee the organizers will cover their ticket sales. 32
Name: Date: Period: Algebra I Modeling Inequalities 2B F Homework 1) Translate each of the following phrases into an inequality, then find the solution set by solving the inequality. (a) When 4 times a number is decreased by 3, (b) When 6 less than 3 times a number is increased it s at most 21. by 2, it s at least 5 times the same number decreased by 8. (c) Find all numbers such that a third of a number (d) The sum of 2 consecutive integers is at most the increased by half that number is at least 3 less than difference between nine times the smaller and 5 that same number. 5 times the larger. (e) The sum of two consecutive even integers is at most seven more than half the sum of the next two consecutive even integers. (f) A fish tank can hold at most 315 gallons of water. If a hose is filling the fish tank at a rate of 15 gallons every 10 minutes, how many hours can the hose be left on before the tank overflows? 33
2) At an amusement park there s only enough room for 4,500 people to be in it at any time. The manager has also worked out that there needs to be 2,800 people in the park to make a profit after all the overhead costs and employee pay. If people are entering the park at a rate of 12 people a minute and there are 850 people in the park currently between how many minutes should the door stay open to let guests in? (a) Translate the scenario above into a compound inequality involving the number of minutes,, that the door has been open. Take into account both the fact that there must be a minimum of 2,800 people and a maximum of 4,500 people. (Hint: This is an And compound inequality) (b) Solve the compound inequality you have stated from part (a). (c) Write the solution set as a single statement in interval notation. 34
Review Section: 3) 4) 5) Find the product of and 35