# Solution Sets for Equations and Inequalities

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Common Core For Algebra I

Examples, solutions, and videos to help Algebra I students understand that an equation with variables is often viewed as a question asking for the set of values one can assign to the variables of the equation to make the equation a true statement.

### New York State Common Core Math Algebra I, Module 1, Lesson 11

Worksheets for Algebra 1

Student Outcomes

Students understand that an equation with variables is often viewed as a question asking for the set of values one can assign to the variables of the equation to make the equation a true statement. They see the equation as a “filter” that sifts through all numbers in the domain of the variables, sorting those numbers into two disjoint sets: the Solution Set and the set of numbers for which the equation is false.

Students understand the commutative, associate, and distributive properties as identities, e.g., equations whose solution sets are the set of all values in the domain of the variables.

Definitions

The solution set of an equation written with only one variable symbol is the set of all values one can assign to that variable to make the equation a true number sentence. Any one of those values is said to be a solution to the equation.

To solve an equation means to find the solution set for that equation.

One can describe a solution set in any of the following ways:

In Words: a2 = 25 has solutions 5 and -5. (That is, a2 = 25 is true when a = 5 or a = -5.)

In Set Notation: The solution set of a2 = 25 is {5, -5}.

It is awkward to express the set of infinitely many numbers in set notation. In these cases we can use the notation: {variable symbol number typ e| description}. For example {x real | x > 0} reads, “x is a real number where x is greater than zero". The symbol can be used to indicate all real numbers.

In a Graphical Representation on a Number Line: In this graphical representation, a solid dot is used to indicate a point on the number line that is to be included in the solution set. (WARNING: The dot one physically draws is larger than the point it represents! One hopes that it is clear from the context of the diagram which point each dot refers to.)

An identity is an equation that is always true.
An equation is an identity when both sides are equivalent.

Exit Ticket

1. Here is the graphical representation of a set of real numbers:

a. Describe this set of real numbers in words.

b. Describe this set of real numbers in set notation.

c. Write an equation or an inequality which has the set above as its solution set.

2. Indicate whether each of the following equations is sure to have a solution set of all real numbers. Explain your answers for each.

a. 3(x + 1) = 3x + 1

b. x + 2 = 2 + x

c. 4x(x + 1) = 4x + 4x2

d. 3x(4x)(2x) = 72x3

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