Solving Equations That Have Symbols of Inclusion

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

Many mathematical equations involve symbols of inclusion. This lesson will define symbols of inclusion, show how to simplify expressions with these symbols, and explain the steps involved in solving equations with symbols of inclusion.

Symbols of Inclusion

Suppose you and some of your friends make up a singing group, to which you are looking to add members. You hold auditions for two days. After the first day, you've added three members to your group and on the second day, you double the number of members you had after the first day.

Here's some interesting mathematical information (come on, you knew it was coming)! If we let x be the number of members in the group before tryouts, then after the first day, there are x + 3 members. Then on the second day of auditions, this is doubled, so there are 2(x + 3) members in the singing group, where x is the number of original members.

In mathematics, we call the parentheses within the expression 2(x + 3) symbols of inclusion. Symbols of inclusion are symbols used in mathematical expressions that group terms or factors together. They indicate that when we are simplifying expressions, we are to perform what's inside the symbols first.

There are three main types of symbols of inclusion. Those are parentheses, brackets, and braces. The order in which we simplify expressions with these symbols is parentheses first, brackets second, and braces third.


When we have a variable inside a symbol of inclusion, we can eliminate the symbols using distribution. To do this, we simply distribute the factor on the outside of the symbol by multiplying it by everything inside the symbol. For example, suppose we want to simplify the following expression:

3[4 + 5(x - 1)]

We would eliminate the symbols of inclusion in the proper order using distribution. Starting with the parentheses, we distribute the 5 by multiplying it by x and -1.

3[4 + 5x - 5]

Now, we simplify within the brackets.

3[5x - 1]

Lastly, we distribute the 3 by multiplying it by 5x and -1.

15x - 3

This process is good to be familiar with because it is key to solving equations that have symbols of inclusion. Speaking of which, let's talk about that!

Solving Equations With Symbols of Inclusion

Let's go back to your singing group. We said that the number of members after tryouts can be represented with the expression 2(x + 3), where x is the number of original members in the group. Suppose that after tryouts, there are 12 members in the group and we want to know how many original members there were. Well, we have that there are 2(x + 3) members after tryouts and we know there are 12 members after tryouts, so it must be that:

2(x + 3) = 12

Perfect! If we can solve this equation for x, we will have the number of original members of the group. All we have to do is figure out how to solve the equation.

Because the equation contains symbols of inclusion, we call it an equation with symbols of inclusions. The steps we want to follow to solve these types of equations are as follows:

  1. Use distribution to eliminate symbols of inclusion.
  2. The resulting equation doesn't contain symbols of inclusion, so solve it as you normally would by isolating the variable on one side of the equation.

Two steps? That's not too bad! Let's give it a go with our equation.

First, we eliminate the parentheses using distribution by distributing the 2.


Now, we have the equation 2x + 6 = 12, so we move on to step 2 and solve the equation for x.


We end up with x = 3, so there were three original members of your singing group. Let's consider one more example to help us get used to this solving process.

Another Example

Let's use our steps to solve the following equation:

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