Introduction to inequalities

So far we have learnt about equations, where one side is equivalent (or equal to) the other side. But there will be times when we want to compare two quantities where one is smaller or larger than the other. To understand inequalities, let's first recap equality.

Equality means that two quantities are equal in value and their equation is balanced. This is indicated by the equals sign in equations. For example, we can write statements such as $1=1$1=1, $1+2=3$1+2=3 and $5=x$5=x.

An inequality exists when one amount is not equal to the other. That is, when one side of the expression is greater than the other. We can think of this like a set of unbalanced scales, where one side is heavier than the other.

 

Inequality Symbols

In mathematics we have special symbols to indicate that an inequality exists. Let's run through them now.

The greater than symbol $>$> indicates that the expression on the left of the symbol has a larger value than the expression on the right of the symbol. For example, $3>2$3>2 means "$3$3 is greater than $2$2".

Similarly, the less than symbol $<$< indicates that the expression on the left of the symbol has a smaller value than the expression on the right of the symbol. For example, $3<4$3<4 means "$3$3 is less than $4$4".

The greater than or equal to symbol $\ge$≥ indicates that the expression on the left of the symbol has a larger value than or is equal to the expression on the right of the symbol. So we could write $3\ge2$3≥2, as well as $2\ge2$2≥2, or $n\ge2$n≥2. This last expression "$x$x is greater than or equal to $2$2" is true for a whole range of values of $n$n, such as $n=2$n=2, $n=2.5$n=2.5 and $n=11$n=11.

Similarly, the less than or equal to symbol $\le$≤ indicates that the expression on the left of the symbol has a smaller value than or is equal to the expression on the right of the symbol. So we could write $3\le4$3≤4, as well as $4\le4$4≤4, or $x\le4$x≤4. Once again, the last expression "$x$x is less than or equal to $4$4" is true for a whole range of values of $x$x, such as $x=4$x=4, $x=3.5$x=3.5 and $x=-1$x=−1.

 

The images below show another demonstration of the inequality symbols:  

 

Worked Examples

Question 1

Solve: Write a mathematical statement to mean "two is greater than one".

Think: The words "greater than" refer to the symbol $>$>.

Do: We can write this statement as $2>1$2>1.

Question 2

Solve: Write a mathematical statement to mean "$k$k is less than or equal to seven".

Think: The words "less than or equal to" refer to the symbol $\le$≤.

Do: We can write this statement as $k\le7$k≤7.

Reflect: There are a whole range of values of $k$k which make this inequality true, including $k=7$k=7, $k=6.998$k=6.998, $k=0$k=0 and $k=-15.5$k=−15.5.

 

Reversing an Inequality

We are familiar with being able to write an equation in two orders. For example, $x=10$x=10 and $10=x$10=x mean the same thing.

We can also write inequality statements in two orders, but we now need to be careful and switch the inequality sign being used as well. For example, $x>10$x>10 means the same thing as $1010<x. That is, "$x$x is greater than ten" is the same as "ten is less than $x$x".

 

Practice Questions

Question 1

Question 2

Question 3