Explainer: Introduction to Inequalities

In this explainer, we will learn how to interpret and write simple inequalities to represent verbal expressions.

Inequalities have something common with equations: they give us information about the relationship between two numbers or expressions (which can include variables). The difference is, however, that this relationship is not an equality, as the name suggests. An inequality states that the value of one expression is greater or less than that of another.

While, in an equation, the equal sign is used between the two expressions, there are four different symbols that can be used for inequalities.

Symbols for Inequalities

  • π‘Ž<𝑏 means that π‘Ž is less than 𝑏.
  • π‘Ž>𝑏 means that π‘Ž is greater than 𝑏.
  • ≀ combines < and =βˆΆπ‘Žβ‰€π‘ means that π‘Ž is less than or equal to 𝑏.
  • β‰₯ combines > and =βˆΆπ‘Žβ‰₯𝑏 means that π‘Ž is greater than or equal to 𝑏.

Let us look at the first question to see how we identify an inequality.

Example 1: Identifying an Inequality

What can you say about π‘₯>27βˆ’6?

  1. It is neither an equation nor an inequality.
  2. It is an equation.
  3. It is an inequality.

Answer

We have here a mathematical sentence, π‘₯>27βˆ’6, and we need to identify whether it is an equation, an inequality, or neither. We see that the symbol > is used; hence, it is an inequality.

Now, we are going to see with the next questions how to write inequalities. Let us start with a very simple example.

Example 2: Writing a Simple Inequality

Write an expression for 𝑐 which means that it is less than or equal to 3.

  1. 𝑐<3
  2. 𝑐>3
  3. 𝑐β‰₯3
  4. 𝑐≀3

Answer

The symbol used to say β€œless than or equal to” is ≀; hence, we write that 𝑐 is less than or equal to 3 with the expression 𝑐≀3.

Option D is the right answer.

We are going now to learn how to write an inequality to represent a real-life situation. We start with a simple situation.

Example 3: Writing an Inequality to Represent a Real-Life Situation

Write an inequality which describes the following sentence: The subscription fee is no more than $74.

  1. π‘₯<74
  2. π‘₯>74
  3. π‘₯β‰₯74
  4. π‘₯≀74

Answer

When we say that something is β€œno more than” $74, this means that it cannot exceed $74. So, it is either equal to $74 or less than $74. Here we are talking about the subscription fee. If we represent the subscription fee by π‘₯, we can write π‘₯≀74,

which means π‘₯ is less than or equal to 74.

We are going to look now at two more complex examples. For these, we need to read carefully the question to understand all the information and be able to translate it into a mathematical sentence. We will see how drawing a simple diagram can help us with this.

Example 4: Writing an Inequality to Represent a Real-Life Situation

Matthew ran 600 meters in 90 seconds, while Natalie was at least 9 seconds ahead of him. Write an inequality to represent the time Natalie finished the 600-meter run.

  1. π‘₯≀99
  2. π‘₯β‰₯81
  3. π‘₯β‰₯99
  4. π‘₯≀81
  5. π‘₯<90

Answer

In this story, we need to visualize Matthew and Natalie running the race; Natalie is ahead, so she arrives before Matthew. Therefore, she ran the race in a shorter time. It is said that she was at least 9 seconds ahead. The β€œat least” means that the time she was ahead was 9 seconds or more.

We can draw a diagram to help us visualize the situation.

We see that Natalie’s time is given by subtracting the β€œ9 seconds or more” from Matthew’s time (90 seconds). If the time she was ahead was exactly 9 seconds, then her time is 90βˆ’9=81s.

If the time she was ahead was more than 9 seconds, then we need to subtract more than 9 s from 90 s, so we get a time shorter than 81 seconds. We see that her time is less than or equal to 81 seconds. Let π‘₯ be her time in seconds. We can write this statement as π‘₯≀81.

Example 5: Writing an Inequality to Represent a Real-World Situation

Scarlett needs to travel by train, and she’s preparing her journey to the train station. It takes her 7 minutes to walk to the bus stop from home. There is a bus every 12 minutes, and the bus journey to the train station usually lasts 20 minutes. After that, she needs at most 4 minutes to get off the bus and reach her train.

Assuming her bus is not delayed, write an inequality to describe the time, 𝑑, in minutes, it takes her to travel to the train station by bus.

  1. 𝑑≀43
  2. 𝑑≀35
  3. 𝑑β‰₯43
  4. 𝑑<43

Given that her train departs at 14:47, at what time should she leave home to make sure that she catches the train?

  1. Before 14:04
  2. After 14:04
  3. At 14:16
  4. After 14:16
  5. Between 14:04 and 14:16

Answer

We need to figure out the time it takes Scarlett to travel to the train station by bus. We know her journey involves a 7-minute walk and a 20-minute bus journey. But she may have to wait for the bus. We are told there is a bus every 12 minutes. This means that if she has just missed the bus, she will have to wait for 12 minutes; this is the maximum waiting time. So, the time she may wait for the bus is no more than 12 minutes: it is less than or equal to 12 minutes. And, finally, we know that she needs no more than 4 minutes to walk from the bus stop to the train platform.

Let us represent all this on a time line.

We see here that 7+12+20+4=43 gives the maximum time in minutes, 𝑑, it takes her to travel from her place to the train platform. This means that 𝑑 is less than or equal to 43 minutes, which is written as 𝑑≀43.

In the next question, we want to find at what time she should leave her place if her train departs at 14:47. She does not want to take any risk missing her train, so she should consider she might need 43 minutes to travel. So, she should leave no later than 43 minutes before the train departs. The train departs at 14:47. And 43 minutes before 14:47 gives us 14:04. Therefore, she should leave no later than at 14:04, which can also be expressed as β€œat the latest at 14:04” or simply β€œbefore 14:04.”

The correct answer is therefore Option A β€œbefore 14:04.”

Key Points

  1. An inequality is a mathematical sentence stating that the value of one expression (or number) is less or greater than the value of another expression (or number).
  2. There are four symbols that can be used to write an inequality:
    • π‘Ž<𝑏 means that π‘Ž is less than 𝑏;
    • π‘Ž>𝑏 means that π‘Ž is greater than 𝑏;
    • π‘Žβ‰€π‘ means that π‘Ž is less than or equal to 𝑏;
    • π‘Žβ‰₯𝑏 means that π‘Ž is greater than or equal to 𝑏.
  3. When writing an inequality to describe a real-life situation, read carefully the question to understand all the information and be able to translate it into a mathematical sentence.
  4. Draw a simple diagram to help you understand the information given and find how to use it.

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