2.08 Solving multi-step inequalities

Solving multi-step inequalities

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 simple inequalities. We are 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.

 

Graphing solutions on a number line

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):

 

 

 

Practice questions

Question 1

Question 2

Question 3

 

Problem-solving with inequalities

We have now looked at solving inequalities that involve one or two steps to solve. We're now going to take a look at how we can use inequalities to solve problems given a written description.

Much as with solving equations from written descriptions, there are certain keywords or phrases to look out for. When it comes to inequalities, we now have a few extra keywords and phrases to represent the different inequality symbols.

 

Worked example

Question 4

Construct and solve an inequality for the following situation:

"The sum of $2$2 lots of $x$x and $1$1 is at least $7$7."

Think: "At least" means the same as "greater than or equal to". Also "lots of" means there is a multiplication, and "sum" means there is an addition.

Do: $2$2 lots of $x$x is $2x$2x, and the sum of this and $1$1 is $2x+1$2x+1. So altogether we have that "the sum of $2$2 lots of $x$x and $1$1 is at least $7$7" can be written as $2x+1\ge7$2x+1≥7.

We can now solve the inequality for $x$x:

$2x+1$2x+1	$\ge$≥	$7$7
$2x$2x	$\ge$≥	$6$6
$x$x	$\ge$≥	$3$3

So the possible values of $x$x are those that are greater than or equal to $3$3.

 

Practice questions

Question 5

Question 6

Question 7

 

Identifying solutions to linear inequalities

So far we have looked at how to solve inequalities for all of the possible solutions, and we found that the process was very similar to how we solve equations.

We don't always need to solve for all of the possible solutions, however. Sometimes we just want to know if a particular value will satisfy an inequality or not.

 

Testing values

Think about the inequality given by $p-7<12$p−7<12. There is an interval of values that $p$p can take to make the inequality true. 

For example, we can test whether $p=20$p=20 satisfies $p-7<12$p−7<12 by simply substituting that value into it and see what happens.

So, if $p=20$p=20, the expression becomes $20-7<12$20−7<12 or, in other words, $13<12$13<12 which is clearly not true. This means that $p=20$p=20 is not in the solution set. It doesn't satisfy the inequality.

On the other hand, a number like $p=10$p=10 is in the solution set because $10-7<12$10−7<12.

 

Worked examples

QUESTION 8

Solve: Does $t=6$t=6 satisfy the equation $3t-5\ge12$3t−5≥12?

Think: We can simply substitute $t=6$t=6 to find out.

Do: So when $t=6$t=6 we have:

$3\times6-5$3×6−5	$\ge$≥	$12$12
$18-5$18−5	$\ge$≥	$12$12
$13$13	$\ge$≥	$12$12

So the answer is yes, $t=6$t=6 does satisfy the inequality.

It's also possible to apply this method to a practical situation.  See the practice question below for an example.

 

Practice question

Question 9