KS3

Inequalities

Part of MathsInequalities

Key points

An inequality - n is less than two – the symbol for less than is highlighted orange. Underneath, a number line increasing in ones from left to right from minus one to five labelled n. A circle is above two on the number line – it is not shaded. An arrowed line points to the left from the circle across the number line – it is highlighted orange. — Image caption,
A number line can be used to illustrate an inequality.

Inequalities are used to describe the relationship between expressions that are not equal. They can be presented algebraically, in words, or on a number line.
The two sides in an are equal. This is not the case in an inequality, as one side may be greater than (or less than) the other.
An inequality can have multiple solutions. When a single inequality is used there can be an number of solutions. In some cases, the number of solutions can be . For example, when the unknown value is an and between two given values.
Solutions to an inequality may be given as a simplified inequality, a finite list of values, a or as an illustration on a number line.

An inequality - n is less than two – the symbol for less than is highlighted orange. Underneath, a number line increasing in ones from left to right from minus one to five labelled n. A circle is above two on the number line – it is not shaded. An arrowed line points to the left from the circle across the number line – it is highlighted orange. — Image caption,
A number line can be used to illustrate an inequality.

Video

Watch the video to learn how Shahan, a video games developer, uses inequalities in his work and find out why inequalities are important when it comes to designing new video games.

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Understanding and using inequality symbols

Inequality symbols are used when values are not equal.

The used when writing inequalities is: ≠, <, ≤, > or ≥, where:

≠ means ‘not equal to’
< means ‘less than’
> means ‘greater than’
≤ means ‘less than or equal to’
≥ means ‘greater than or equal to’

Examples

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The inequality symbols are used when values are not equal.
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When each side of the inequality are not equal the ‘not equal to’ symbol may be used. 4 is not equal to 9. The sum (1 + 2 = 3) is not equal to the difference (2 – 1 = 1). The ‘not equal to’ symbol does not order the values, it just states that they are not the same.
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The ‘less than’ and ‘greater than’ symbols give a comparison. The smaller end always points towards the lesser value. The bigger end opens out to the greater value.
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𝒏 < 4 means 𝒏 is less than 4. The value of 𝒏 is any number that is smaller than 4. This could be 3, 2, 1, 0, – 1, – 2 and so on for integers. The value of 𝒏 can also be a decimal or fraction, such as, 3۰9, 2۰15, and ¾ . The symbol ‘less than or equal to’ means that 𝒏 can also have the value that is given. In this example, 𝒏 could be 4 as well as any value less than 4. For both statements there are an infinite number of values that 𝒏 can take.
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𝒏 > 4 means 𝒏 is greater than 4. The value of 𝒏 is any number that is bigger than 4. This could be 5, 6, 7, 8, and so on for integers. The value of 𝒏 can also be a decimal or fraction, such as 4۰1, 5¾, and 9۰02. The symbol greater than or equal to means that 𝒏 can also have the value that is given. In this example, 𝒏 could be 4 as well as any value greater than 4. For both statements there are an infinite number of values that 𝒏 can take.
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The sum 1 + 2 is not equal to the difference 2 – 1. 3 is not equal to 1. Working out the answer to each expression checks whether the inequality statement is correct. The inequality can also be written as 1 + 2 > 2 – 1 because 3 > 1 (3 is greater than 1).
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The expression on each side of the inequality is worked out to check that the inequality is true. 3 – 7 = –4 and –4 is greater than 5. This means 3 – 7 > 5 is false. The true inequality is 3 – 7 < 5. The corrected inequality states that 3 – 7 is less than 5
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The words used to describe an inequality can vary. For example, ‘𝒂 is greater than or equal to 𝒃’ can also be described as ‘𝒂 is at least 𝒃’.
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When the variable is given between two inequalities and values, it may be possible to list the integer solutions. In this example, 𝒏 is the integer. 3 is less than 𝒏, which is less than or equal to 6. 𝒏 can be any integer that is greater than 3, up to and including 6. The set of integer solutions is {4, 5, 6}.

Question

Which inequality is used to state that 𝒏 is at least 8?

Inequalities on a number line and listing integer solutions

Inequalities can be illustrated on a .

For a single inequality, with one value:

Draw a circle above the value on the number line.
- The circle is unshaded for < and >
- The circle is shaded for ≤ and ≥
Draw a horizontal line with an arrow in the same direction as the inequality:
- to the left with an arrow for < and ≤
- to the right with an arrow to the right for > and ≥

For a single inequality there are an infinite number of integer solutions. These can’t be listed in a solution set.

For an inequality with two values:

Draw circles above the values on the number line.
- The circle is unshaded for < and >
- The circle is shaded for ≤ and ≥
Draw a horizontal line between the circles.

For an inequality with two values there are a finite number of integer solutions. This can be written as a solution set which is presented in curly brackets. For example, {2, 3, 4}.

Examples

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A number line can be used to illustrate an inequality.
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Show the inequality 𝒏 < 2 on a number line.
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The inequality uses the value 2. Draw a circle above the value 2 on the number line. The number line must show the value/s in the inequality and does not need to be a specific length. The circle is not shaded because 𝒏 is less than 2. 𝒏 cannot be 2
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To show that 𝒏 is less than 2, draw a horizontal line with an arrow pointing to the left (to lesser values). The arrowed line does not have to be a particular length, it just needs to show that 𝒏 can have values that are less than 2. The arrow points in the same direction as the inequality. The integer solutions for the inequality cannot be listed as a solution set because an infinite number of integers are possible.
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Show the inequality 𝒏 ≥ 1 on a number line.
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The inequality uses the value 1. Draw a circle above the value 1 on the number line. The circle is shaded because 𝒏 can have the value 1
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To show that 𝒏 can be greater than 1, draw a horizontal line with an arrow pointing to the right (to greater values). The arrowed line does not have to be a particular length, it just needs to show that 𝒏 can have values that are more than 2. The arrow points in the same direction as the inequality. The integer solutions for the inequality cannot be listed as a solution set because an infinite number of integers are possible.
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Draw the inequality 2 ≤ 𝒏 < 5 on a number line.
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The inequality uses the values 2 and 5. A shaded circle is above the value 2 on the number line, because 𝒏 can have the value 2. An unshaded circle is drawn above the value 5 on the number line, because 𝒏 cannot have the value 5
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Draw a horizontal line between the two circles to show that 𝒏 can take any value between the two numbers. The integer solutions for the inequality can be listed as a solution set because a finite number of integers are possible. In this example, 𝒏 can be 2 (solid circle), 3 or 4. 5 is not included, because the circle is unshaded. The solution set is {2, 3, 4}. The values of a solution set are listed within curly brackets.

Question

What inequality is shown on the number line?

A number line increasing from left to right in ones from minus one to three – this is labelled n. A circle is above zero – it is unshaded. A circle is above three – it is unshaded. A line joins the two circles.

Solving simple linear inequalities

A single inequality written with an may be solved by using :

Undo each process in the inequality by using the inverse operation. Inverse operations include addition and subtraction, and multiplication and division.
Start with the last operation and work back to the first.
For each step make sure that the inverse is applied to both sides of the inequality.
Once the variable is write the solution as a simple inequality.

An expression between inequalities is solved using inverse operations in the same way:

Apply each inverse operation to all three parts of the inequality.
Write the solution as a variable between two inequalities.

Examples

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Solve the inequality 3𝒏 ≥ 24
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The variable (𝒏) has been multiplied by 3 to give a value that is greater or equal to 24. The inverse of × 3 is ÷ 3
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Because the inverse of × 3 is ÷ 3, divide both sides of the inequality by 3. This gives 𝒏 ≥ 8. The 𝒏 is isolated. The solution to the inequality is 𝒏 is greater or equal to 8
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Solve the inequality 2𝒏 – 1 < 9. The variable (𝒏) has been multiplied by 2. 1 is then subtracted to give a value that is less than 9. The inverse of × 2 is ÷ 2. The inverse of – 1 is + 1
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The inverse of – 1 is + 1. Add 1 to both sides of the inequality. The inequality is now 2𝒏 < 10
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The inverse of × 2 is ÷ 2. Divide both sides of the inequality by 2. The 𝒏 is isolated. The solution is n < 5
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𝒏 is an integer. Find the integer solutions for 7 ≤ ⁿ⁄₂ + 4 < 9
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The variable (𝒏) has been divided by 2. 4 is then added to give a value that is greater than or equal to 7 and is less than 9. The inverse of ÷ 2 is × 2. The inverse of + 4 is – 4
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The inverse of + 4 is – 4. Subtract 4 from each part of the inequality. The inverse of ÷ 2 is × 2. Multiply each part of the inequality by 2. The variable 𝒏 is now isolated. This makes the solution 6 ≤ 𝒏 < 10
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The integer solutions for 6 ≤ 𝒏 < 10 are all the integer values that 𝒏 can be. The least value 𝒏 can be is 6, because 6 is less or equal to 𝒏. 𝒏 cannot be 10, because 𝒏 is less than 10. The integer values for 𝒏 can be 6, 7, 8 or 9. The solution set for 7 ≤ ⁿ⁄₂ + 4 < 9 is {6, 7, 8, 9}.

Question

Solve the inequality and list the integer solutions for 𝒏

An inequality. Eighteen is less than seven n minus ten is less than or equal to twenty five.

Practise using inequalities

Quiz

Game - Divided Islands