Topics 5: Inequalities and Regions

By the end of this topic, you will be able to.

1. identify and use Inequality symbols
2. Illustrate Inequalities on the number lines,
3. Solve linear inequalities in one unknown.
4. represent linear inequalities graphically.
5. form s.rnple linear inequalities tot
   regions on a graph.

Keywords

• at least
• at most
• greater than equal to
• inequality
• than
• less than or equal to
• more than

Introduction

Have you ever saved up money to buy something you really want? You can determine if you have enough money by counting the money you have saved and comparing that amount to the cost of the item you want ‘o buy When we compare two numbers that are equal we can the comparison inequality. Therefore by end of the topic. you Will be able to represent and solve problems Involving Inequalities

5.1 Identifying and using inequality symbols

In this section. you learn about Inequalities. Including the symbols used In writing inequalities, writing inequalities using variables and row Inequalities Can be useful In real work You will find symbols such as c, s. which all have different meanings

Activity 5. Using inequality symbols
Begin by talking about buildings •n your town, or a city nearby.

1. List any tall buildings in the picture.
2. Among the buildings listed, are there any of the same height? If so, list them.
3. What mathematical statement would you use to describe the buildings listed in (2)?
4. Are there any buildings taller than the others?
5. List the buildings and use a mathematical symbol to describe
   the buildings listed in (4).
6. Are there any buildings shorter than the others?
7. List the buildings and use a mathematical symbol to describe
   the buildings listed in (6).

Activity 5.2 Forming inequalities from word problems

Jacqueline wanted to buy the newest movie from a video library in the trading centre She found that the movie costs UGX 2.000, She had
saved cons worth UGX 1.000 In her savings box.
1. Does Jacqueline have enough money to buy the movie? Give an explanation for your answer-
2. Write a number statement to represent the amount of money Jaqueline has saved in comparison to the cost of a movie at the video library In the trading centre.
3. How would you read the number statement in (2)?

Activity 5.3 Finding out more about Inequalities
The Uganda Mathematical Society set up a Mathematics Junior contest
for Semor Two. It was marked out to a total of 50. The Junor contest
was meant to sort out those who would qualify for the national
mathematics contest.

Learners who would score at least 35 were to merit the national
mathematics contest paper The learners who didn’t get more than 25
were eliminated from the contest and the remainders were to be given
a second test to qualify them for the national mathematics contest.

1. Write the range of those who passed on merit.
2. Write the range of those who were eliminated.
3. Write the range of those who were to be given a second test.

Learning points
Words such as “more than, greater than”, taller than, less than shorter than/fewer than e.t.c are used to express inequality. Each of these phrases; more than/greater than and less than has a mathematical symbol used to describe it.

Exercise 5.1
1. Write down the meaning of each of the following inequalities. Meaning

2. Write inequalities that describe each of the following statements.
(a) Martin weighs more than Marvin.
(b) An elephant is bigger than a dog.
(c) A pencil is cheaper than a pen.
(d) The speed limit on a certain road is 60 km.
(e) The area of the circle is……. the area of the triangle.

3. Rewrite the following using inequality symbols.
(a) t is less than 6.
(b) t is greater than or equal to 9.
(c) t is more than 23.
(d) p is greater than 6 and less than 8.
(e) y is more than or equal to 5.
(f) x is less than or equal to 15.
(g) x is more than or equal to 4 but less than or equal to 8.

4. Write the following inequalities in words.
(a) y <3
(d) 8 sys 12
(b) ts8
(c) t>24

5.2 Illustrating inequalities on the number lines

In this section, you will explore linear inequalities and make connections among multiple representations (including algebraic expressions, verbal statements, number line graphs, and solution sets). At primary school, you learnt about a number line and directed numbers.

Activity 5.4 Illustrating Inequalities on the Number Lines

1. Write down the inequalities represented on these number lines.
   (a)

2. What are the solutions represented by the inequalities written in (1)?

3. How would you relate the solutions written in (2) with the dots used in each of the number lines?

Learning points

• When you are working with inequalities, a common question to be asked is to state all the numbers that satisfy an inequality. One can either do this by giving a solution set or by drawing the numbers on a number line.
• For the inequality greater than/less than sign, conventionally use an unshaded circle/empty circle. The unshaded/empty dot means that the number is not part of the solution set.
• For the inequality less than/greater than or equal to sign, conventionally use a shaded circle/solid dot. The shaded circle or solid dot means the number is inclusive of the solution set.

5.3 Solving linear inequalities in one variable

In this section, you are going to learn that linear inequalities give a set of solutions as opposed to just one solution. You can solve linear inequalities using similar methods as in solving multi-step equations, except that there are extra rules when using multiplication and division.

Activity 5.5 Solving linear inequalities in one variable

1. Sarah multiplied both sides of the inequality 1 <5 by -1. Her answer was -1 <-5.
   (a) Is she correct?
   (b) Explain and show how you obtained the correct answer in (a).
2. The table below is based on a number line from which you are to choose values C and D depending on the description given. These values are affected by P, as shown in the first column. Fill in the rest of the table correctly.

3. (a) Solve the inequalities: x+5<10, 2x+5<-15, -2x+5<15.

(b) What do you learn from the solutions of these different inequalities?

Learning points

• Inequalities follow different rules from those used in equations when dealing with multiplication.
• Solving inequalities using multiplication involves watching for negative numbers.
• When we multiply both sides by a negative number, we must change the direction of the inequality sign, e.g. 1 < 5 becomes -1 > -5. This happens when using multiplicative inverses to simplify the way we would when solving multi-step equations.

Exercise 5.3

1. Write an inequality comparing the two given numbers. Then complete the table using what you have learnt about working with inequalities.

2. Solve the following inequalities for xER. In each case, represent the solution set on a number line.
(a) x+3<5 (c) 2x+3<5 (e) 12x + 3 ≥ 11 3. 4. (b) X-6<5 (d) 2x + 3 < 15

3. Which of the following is an element of the solution set of the inequality 4x + 5 > 29?
(I) 4
(ii) -5
(ii) 6
(iv) 7

4. Martin worked a six-hour shift in his local restaurant and got UGX 22,500 in tips. His total take-home pay that evening was at least UGX 310,500. Find the minimum amount he was paid per hour.

5. A farmer wants to buy some cows and a tractor. The tractor costs UGX 90,000,000 and the maximum the farmer can spend is UGX 270,000,000. Given that the average price of a cow is UGX 4,050,000, find the maximum number of cows the farmer can buy.

6. Sharif is saving for a birthday present for his mother and he already has UGX 27,000. Given that he plans to spend at least UGX 180,000 on the present and the birthday is in five weeks, what is the least amount he should save per week?

5.4 Representing linear inequalities graphically

In this section, you are going to start looking at graphical inequalities. Therefore, you need to recall and know how to plot straight lines.

Activity 5.6 Representing inequalities graphically

1. Which of the following are solutions to the linear inequality y < 4? (3,2), (0,0), (1,1), (2,2), (0,4), (4,0), (2.4), (-1,4), (5,5). Justify your answer.
2. Which of the following are solutions to the linear inequality is 4? (3,2), (0,0), (1,1), (2,2), (0,4), (4,0), (2.4), (-1,4), (5,5). Justify your answer.
3. What can you deduce about the outcomes of the solutions in (1) and (2)
4. Which of the following are solutions to the linear inequality y> 4? (3,2), (0,0), (1,1), (2,2), (0,4), (4,0), (2.4), (-1,4), (5,5). Justify your answer.
5. Which of the following are solutions to the linear inequality y ≥ 4? (3,2), (0,0), (1,1), (2,2), (0,4), (4,0), (2.4), (-1,4), (5,5). Justify your answer.
6. What can you deduce about the outcomes of the solutions in (4) and (5)?
7. With the skills developed from the previous section, plot these inequalities on separate graphs and shade the regions that satisfy
   each of the inequality.
   (a) x+y<4 (c) x + y ≤4 (b) x+y> 4
   (d) x + y ≥4

Learning points

• If you want to solve for or describe a region in a coordinate plane, you can use linear inequalities.
• Graph linear inequalities by shading wanted regions of number lines or coordinate planes.
• Just like on the number line, on the graph; use a dotted line for < and >, and a solid line for > and <.

Example
Show on the graph the following regions and shade the wanted region.
(a) y ≥-8
(b) y < 2x + 3 (c)

Solution
(a)

We draw the line y = -8 which is a solid line to show that values along the line are part of the inequality. Ffor instance, y = -8 and y = 0 satisfy y ≥ 8. Hence the values are in the wanted regions.

We substitute for values of x in the equation, y = 2x + 3 to find corresponding values of y. We draw the line y = 2x + 3, which is a dotted line to show that the values along the line are not part of the inequality. (2,0) do not satisfy y < 2x + 3. For instance, (0, 3).

Thus, the wanted region is the opposite side.

Exercise 5.4 Show on the graph the following regions.

(a) y ≥-8 (b) y < 2x>-x+6 (d) y = 2x-1

4. Abiriga works two part-time jobs. One is at a petrol station that pays UGX 115,500 an hour and the other is IT troubleshooting for UGX 46,200 an hour. Between the two jobs, Abiriga wants to earn at least UGX 1,155,000 a week. How many hours does Abiriga need to work at each job to earn at least UGX 1,155,000?

(a) Let x be the number of hours he works at the petrol station and let y be the number of hours he works troubleshooting. Write an inequality that would model this situation.

(b) Graph the inequality.

(c) Find three ordered pairs that would be solutions to the inequality. Then, explain what that means for Abiriga.

5.5 Forming simple linear inequalities for regions on a graph

In this section, you are going to learn that two or more linear inequalities grouped together form a system of linear inequalities.

Activity 5.7 Solving a system of linear inequalities

1. Determine whether the ordered pair is a solution to the system x + 4y ≥ 10 and 3x – 2y < 12. Explain your answer. (a) (-2,4) (b) (3,1)
2. Graphically represent the linear inequality system by identifying the region that is true for the linear inequality systems in (1).

Learning points

• To solve a system of linear inequalities, you find values of the variables that are solutions to both inequalities.
• Solve the system by using the graphs of each inequality and show the solution on a graph. Find the region on the plane that contains all ordered pairs (x,y) that make both inequalities true.

Exercise 5.5

1. In the following exercises, solve each system by graphing.

(a) y < 3x> x – 1 (b) y < 2x + 2 and y = -x-1

2. Show the region defined by the set of inequalities 3. 4. y < 2x + 5, y ≥ x and x < 4.

3. (a) Draw on the same coordinate axes, lines y + x = 8 and y = 3x.

(b) Use your graph to identify coordinates of points that satisfy both inequalities in order to find regions satisfied by the inequalities x + y ≤ 8 and y ≥ 3x.

4. Lunyolo is studying for her final exams in Chemistry and Mathematics. She knows she only has 24 hours to study, and it will take her at least three times as long to study for Mathematics than Chemistry.

(a) Write a system of inequalities to model this situation.

(b) Graph the system.

(c) Can he spend 4 hours on Chemistry and 20 hours on Mathematics?

(d) Can he spend 6 hours on Chemistry and 18 hours on Mathematics?

Sample Activity of Integration Context Anarika, a university student, has seen three pairs of bedsheets she likes. They cost UGX 225,000, UGX 247,500 and UGX 306,000. She has already saved UGX 90,000 and gets UGX 18,000 pocket money per week at the end of each week. Anarika is wondering how soon she can buy one of these pairs of bedsheets.