topic badge
India
Class XI

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. Make sure to use the correct type of endpoint.

    -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

Outcomes

11.A.LE.1

Linear inequalities. Algebraic solutions of linear inequalities in one variable and their representation on the number line. Graphical solution of linear inequalities in two variables. Solution of system of linear inequalities in two variables – graphically.

What is Mathspace

About Mathspace