Inequalities

Comparison Property

If you are given any two numbers a and b, then there are three possible relationships between them.

Either:

a = b

a  >  b

a  <  b

We are comparing the two numbers, or putting them in order.

To read < and > remember, we read from left to right.

In < the left side is smaller than the right side of the symbol, therefore the symbol is representing less than

In > the left side is bigger than the right side of the symbol, therefore the symbol is representing greater than

An inequality is a mathematical sentence using < or > to compare two expressions.

We can use any of the following inequality symbols to compare numbers:

Symbol

Read as

<

less than

>

greater than

≤

less than or equal to

≥

greater than or equal to

≠

not equal to

Along with =, we can make any sentence true by using the appropriate symbol.

Examples: Replace  with <, >, or = to make the sentence true

1.

-5 ? -7 + 4
-5 ? -3
-5 <  -3
 

Remember; you must simplify first

2.

-2 ? -7 - (-4)
-2 ? -7 + 4
-2 >  -3
 

Remember, a minus sign in front of ( ) changes the -4 to a +4, then add

3.

-7 ? -3 - 4
-7  =  -7

Remember, like signs; add and put that sign

After simplifying expressions, we can use inequality symbols to determine if a sentence is true or false.

Example


Is 8 >  -7 + 12 True or False?

Remember, simplify first:
-7 + 12 = 5

8 >  5

Rule used: unlike signs; subtract and put the sign of the biggest number

Since 8 is greater than 5, the sentence is true.

Graphing Inequalities

We can also use a number line to compare two numbers. When looking at the graphs of any two numbers a and b

  • If the graph of a is to the left of the graph of b then a < b
  • If the graph of a is to the right of the graph of b then a > b

Example

- 4 < 2

The graph of -4 is to the left of the graph of -2

31/4 > 1/2

The graph of 31/4 is to the right of the graph of 1/2

On the previous number line, we graphed 1/2 and 31/4. They are not integers, but instead are examples of rational numbers.

Remember: rational numbers are numbers that can be expressed as a/b where a and b are integers and b ≠ 0.

Integers: {... -3, -2, -1, 0, 1, 2, 3 …} are the numbers we use on a number line.

b ≠ 0 because division by zero is undefined.

So, rational numbers are fractions, or any number that can be written as a fraction.

Example

Rational Number

5

-21/3

.5

0

.333¯

Form a/b

5/1

-7/3

5/10 = 1/2

0/1

1/3

Any integer can be written as a fraction by putting it over the number one (1). Decimals that are terminating (ending) like .5 and decimals that repeat like .333¯ can also be written as fractions and are therefore rational numbers.

We are able to find the solution set of an inequality. Remember, the solution set is any number that makes an open sentence true (an open sentence is a sentence that contains a variable).

Next, we will graph the solution sets of inequalities. To graph inequalities, we use an open circle for < or > or = and a closed circle for ≤ or ≥ .

Example: Graph the solution set of x < 3

Reasoning:

  • We use an open circle for =
  • Our graph is to the left for <
  • We fill in the left arrow because the number line does not end
  • The line appears to be solid because we are graphing rational numbers and not integers

Example: Graph the solution set of x ≥ 0

Reasoning:

  • We use a closed circle because of the =
  • We graph to the right for ≥
  • We fill in the right arrow because the number line does not end
  • We fill in the whole line because we are graphing rational numbers

Writing Inequalities from Graphs

Example: Write an inequality for this graph.

Answer: x ≤ 2

Reasoning:

  • = because of the closed circle
  • ≤ because the graph is to the right

Example: Write an inequality for this graph.

Answer: x < 3

Reasoning:

  • no equal to because of the open circle
  • < because the graph is to the left

Example: Write an inequality for this graph.

There are two possible answers

Answer 1: x < 0

Answer 2: x > 0 or x < 0

Reasoning

  • Open circle at zero means ≠ zero
  • Since both left and right of zero are graphed, left of zero is < and right of zero is >

Using the third example in the last two sets, we need to remember that a = b, a ≥ or a ≤ b. If one of the three possibilities is eliminated, that still leaves the other two possibilities.

Example: x is not greater than -5

Answer: x < -5

Reasoning:

  • x not greater than -5 eliminates = from our answer
  • that leaves both = and <

In Algebra, we refer to positive and negative numbers.

Think: compare to zero!

A positive number is x > 0.

A negative number is x < 0.

Try these exercises:

    Graph each set of numbers on a number line.

  1. {-3, -2, 0, 2, 4}
  2. {4, 5, 6, ...}
  3. {integers less than 3}
  4. {integers between -3 and 2}

    5. Name the set of numbers graphed.

    7. You must recognize and continue the pattern!

    Find the sum on a number line.

  7. -5 + 3 =
  8. 3 + 4 =
  9. -4 + (-2) =
  10. 7 + (-5) =

    11. Identify the next three numbers in the established pattern.

  11. 5, 8, 11,       ,      ,      
  12. 7, 4, 1,       ,      ,      

Answers:

  5. {-3, -1, 1, 3}
  6. {-2, 0, 2, 4, ...}
  7. -5 + 3 = -2

  8. 3 + 4 = 7

  9. -4 + (-2) = -6

  10. 7 + (-5) = 2
  11. 5, 8, 11, 14, 17, 20
  12. 7, 4, 1, -2, -5, -8