Two step inequalities on a number line
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.
Exploration
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):
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.
Practice questions
Question 1
Consider the inequality $3x+1>4$3x+1>4.
Solve the inequality.
Now plot the solutions to the inequality $3x+1>4$3x+1>4 on the number line below.
Question 2
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. Make sure to use the correct type of endpoint.
Question 3
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.