Ontario 10 Academic (MPM2D)
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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.

 

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):

 

 

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.

 

Practice questions

Question 1

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

  1. Solve the inequality.

  2. Now plot the solutions to the inequality $3x+1>4$3x+1>4 on the number line below.

    -10-50510

Question 2

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

  1. Solve the inequality.

  2. Now plot the solutions to the inequality $7-x>13$7x>13 on the number line below.

    -10-50510

Question 3

Consider the inequality $2>2\left(x-5\right)$2>2(x5).

  1. Solve the inequality.

  2. Now plot the solutions to the inequality $2>2\left(x-5\right)$2>2(x5) on the number line below.

    -10-50510

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