Multi-step inequalities

Multi-step inequalities are solved in exactly the same way as are one step inequalities or two step inequalities. The only difference between them is the number of steps you have to perform in order to get to the solution. Since the number of steps in multi-step inequalities is not limited, they can get pretty complicated. But, if you follow the order of operations, remember all you have learned in previous lessons and with a bit of practice as well as logical reasoning, you will have no trouble whatsoever.

We will show you how to deal with these inequalities on this reasonably difficult example:

-8(11n + 6) + 6(8 + 5n) > 2n + n

The first thing that needs to be done is getting rid of the parentheses. We can do that by multiplying the first parentheses on the left with (-8) and the second one with 6. So now we get:

-88n – 48 + 48 + 30n > 2n + n

Now we have to put all the numbers on the right and all the variables on the left side of the expression. Notice that the only numbers we have in this expression are -48 and 48. When we add them together, we get 0 as a result. So now the expression looks like this:

-88n +30n – 2n – n > 0

Now, let us perform the rest of the additions and subtractions:

-61n > 0

We are not done yet. There is still one more step to perform and that is to divide the whole expression by (-61). Since that is a negative number, keep in mind you have to change the sign of inequality to its opposite. So the last step goes like this:

-61n > 0 |: (-61)

n < 0

The final result is every real number less than zero.

multi step inequalities

We hope this lesson helped you and that you were able to learn from it. If you have additional questions or comments, please send them via our contact form and we will try to answer them as soon as possible. If you want to practice solving multi-step inequalities some more, feel free to use the math worksheets below.

Multi-step inequalities exams for teachers

Exam Name File Size Downloads Upload date
Integers
Multi-step inequalities – Integers – very easy 560 kB 3686 October 13, 2012
Multi-step inequalities – Integers – easy 565.4 kB 4189 October 13, 2012
Multi-step inequalities – Integers – medium 586.2 kB 6655 October 13, 2012
Multi-step inequalities – Integers – hard 614.1 kB 3626 October 13, 2012
Multi-step inequalities – Integers – very hard 580.7 kB 3034 October 13, 2012
Decimals
Multi-step inequalities – Decimals – very easy 586 kB 939 October 13, 2012
Multi-step inequalities – Decimals – easy 602.2 kB 984 October 13, 2012
Multi-step inequalities – Decimals – medium 611.7 kB 1193 October 13, 2012
Multi-step inequalities – Decimals – hard 595.7 kB 840 October 13, 2012
Multi-step inequalities – Decimals – very hard 607.4 kB 799 October 13, 2012
Fractions
Multi-step inequalities – Fractions – very easy 569.6 kB 2253 October 13, 2012
Multi-step inequalities – Fractions – easy 586.5 kB 2127 October 13, 2012
Multi-step inequalities – Fractions – medium 576 kB 2106 October 13, 2012
Multi-step inequalities – Fractions – hard 608 kB 1203 October 13, 2012
Multi-step inequalities – Fractions – very hard 585.4 kB 1371 October 13, 2012


Multi-step inequalities worksheets for students

Worksheet Name File Size Downloads Upload date
Multi-step inequalities – Integers 5.9 MB 3654 October 14, 2012
Multi-step inequalities – Decimals 7.2 MB 966 October 14, 2012
Multi-step inequalities – Fractions 6.5 MB 1747 October 14, 2012


Multi-step inequalities knowledge test