Inequalities

UK Secondary (7-11)

Two step inequalities on a number line

Lesson

We have seen that it is easiest to plot an inequality on a number line by first solving the inequality. We have also looked at solving inequalities involving two steps. We're now going to combine these ideas together - let's recap through an example.

Suppose we want to plot the solutions to the inequality $2\left(3+x\right)<8$2(3+`x`)<8 on a number line. That is, we want to plot the values of $x$`x` which can be added to $3$3 and then doubled to result in a number less than $8$8.

To solve this inequality, we want to undo these operations in **reverse order**. That is, we can solve this inequality by first dividing both sides by $2$2, then subtracting $3$3 from both sides:

$2\left(3+x\right)$2(3+x) |
$<$< | $8$8 | ||

$3+x$3+x |
$<$< | $4$4 | Dividing both sides by $2$2 | |

$x$x |
$<$< | $1$1 | Subtracting $3$3 from both sides |

In this case, we arrive at the result $x<1$`x`<1. We can test some values in the original inequality to see if this is the right solution set - let's say $x=0$`x`=0 and $x=2$`x`=2.

- When $x=0$
`x`=0, we have $2\left(3+x\right)=2\left(3+0\right)=6$2(3+`x`)=2(3+0)=6, which**is**less than $8$8. - When $x=2$
`x`=2, we have $2\left(3+x\right)=2\left(3+2\right)=10$2(3+`x`)=2(3+2)=10, which**is not**less than $8$8.

So our result of $x<1$`x`<1 seems to be correct.

We can now plot the solutions on a number line as follows, using a hollow circle for the endpoint (since $x=1$`x`=1 is **not** included in the solutions):

Remember

When solving an inequality:

- Multiplying or dividing both sides by a
**negative**number will reverse the inequality symbol. - It is generally easiest to undo one operation at a time, in reverse order to the order of operations.

When plotting an inequality:

- The symbols $<$< and $>$>
**don't**include the end point, which we show with a**hollow**circle. - The symbols $\ge$≥ and $\le$≤
**do**include the endpoint, which we show with a**filled**circle.

Consider the inequality $3x+1>4$3`x`+1>4.

Solve the inequality.

Now plot the solutions to the inequality $3x+1>4$3

`x`+1>4 on the number line below.

Consider the inequality $7-x>13$7−`x`>13.

Solve the inequality.

Now plot the solutions to the inequality $7-x>13$7−

`x`>13 on the number line below.

Consider the inequality $2>2\left(x-5\right)$2>2(`x`−5).

Solve the inequality.

Now plot the solutions to the inequality $2>2\left(x-5\right)$2>2(

`x`−5) on the number line below.