# An Introduction to Inequalities

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An Introduction to Inequalities
Presented by J. Grossman

Definition An inequality is a mathematical statement that one quantity is greater than or less than another.

≥ : greater than or equal to
An inequality is like an equation, but instead of an equal sign (=) it has one of these signs: < : less than ≤ : less than or equal to > : greater than ≥ : greater than or equal to

A solution of an inequality is any real number that makes the inequality true. For example, the solutions of the inequality x < 3 are all real numbers that are less than 3. Can you name them all? Why? Why not?

x < 5 means that whatever numerical value x has, it must be less than 5. Name ten numbers that are less than 5!

Numbers less than 5 are to the left of 5 on the number line.
5 10 15 -20 -15 -10 -5 -25 20 25 If you said 4, 3, 2, 1, 0, -1, -2, -3, etc., you are right. There are also numbers in between the integers, like 2.5, 1/2, -7.9, etc. The number 5 would not be a correct answer, though, because 5 is not less than 5.

Name ten numbers that are greater than or equal to -2!
x ≥ -2 means that whatever value x has, it must be greater than or equal to -2. Name ten numbers that are greater than or equal to -2!

Numbers greater than -2 are to the right of -2 on the number line.
5 10 15 -20 -15 -10 -5 -25 20 25 -2 If you said -1, 0, 1, 2, 3, 4, 5, etc., you are right. There are also numbers in between the integers, like -1/2, 0.2, 3.1, 5.5, etc. The number -2 would also be a correct answer, because of the phrase, “or equal to”.

Where is -1.5 on the number line? Is it greater or less than -2?
5 10 15 -20 -15 -10 -5 -25 20 25 -2 -1.5 is between -1 and -2. -1 is to the right of -2. So -1.5 is also to the right of -2.

Three Ways to State an Inequality
Use inequality notation. Use interval (or set) notation. Graphically display the solution on a number line. x < –3 x is less than minus three {x | x < –3} all x such that x is less than minus three {x | x is a real number, x < –3} the interval from minus infinity to minus three

Graphing Inequalities
x > c When x is greater than a constant, you darken in the part of the number line that is to the right of the constant. Also, because there is no equal line, we are not including where x is equal to the constant. That means we are not including the endpoint. One way to notate that is to use an open circle at that point on the number line.

Graphing Inequalities
x < c When x is less than a constant, you darken in the part of the number line that is to the left of the constant. Also, because there is no equal line, we are not including where x is equal to the constant. That means we are not including the endpoint. One way to notate that is to use an open circle at that point on the number line.

Graphing Inequalities
x < c When x is less than or equal to a constant, you darken in the part of the number line that is to the left of the constant. Also, because there is an equal line, we are including where x is equal to the constant. That means we are including the endpoint. One way to notate that is to use a closed circle at that point on the number line.

Graphing Inequalities
x > c When x is greater than or equal to a constant, you darken in the part of the number line that is to the right of the constant. Also, because there is an equal line, we are including where x is equal to the constant. That means we are including the endpoint. One way to notate that is to use a closed circle at that point on the number line.

Practice Graphing Inequalities

Practice Graphing Inequalities
x is less than 5 y is greater than -3 A number n is positive b is less than or equal to eight The speed limit is posted 55 mph. Write and graph an inequality for this situation.

More practice… state the inequality that has been graphed.

From the Internet SpeedMath -- Inequalities
Online Graphing Calculators

Any Questions??? Homework… Pre-Algebra Study Guide Practice 2-8,
all problems. Algebra Study Guide Practice 3-1,

Next step... Solve Inequalities Using Addition or Subtraction
Using Multiplication or Division

Reference: CPM Algebra Connections

Solving an Inequality Using Addition or Subtraction
Solving inequalities that involve addition or subtraction is just like solving equations that involve addition or subtraction. When solving linear inequalities, we use a lot of the same concepts that we use when solving linear equations.  Basically, we still want to get the variable on one side and everything else on the other side by using inverse operations.  The difference is, when a variable is set equal to one number, that number is the only solution.  But, when a variable is less than or greater than a number, there are an infinite number of values that would be a part of the answer.

Solving Inequality Using Addition or Subtraction
Example 1: Solve the following inequality and graph the solution set. Graph:

Solving an Inequality Using Addition or Subtraction
Example 2: Solve the following inequality and graph the solution set.                 Graph:

Solving an Inequality Using Addition or Subtraction
Example 3: Solve the following inequality and graph the solution set.                Graph:

Solving an Inequality Using Addition or Subtraction
Example 4: Solve the following inequality and graph the solution set.

Solving an Inequality Using Addition or Subtraction
Example 5: Solve the following inequality and graph the solution set. 4x > 28 A. {x | x < 4} B. {x | x < 7} C. {x | x > 28} D. {x | x > 7}

Solving an Inequality Using Addition or Subtraction
Addition/Subtraction Property for Inequalities If a < b, then a + c < b + c If a < b, then a - c < b – c In other words, adding or subtracting the same expression to both sides of an inequality does not change the inequality.