Let us try with x = 6,

5 + 6 < 9

this is equivalent to

11< 9       which is false.

 

In conclusion: any number grater and equal to 6 is not a solution, so we need to look for smaller numbers

Let us try with x = 3,

5 + 3 < 9

this is equivalent to

8 < 9         which is true.

 

In conclusion: any number less and equal to 3 is a solution

 

 

 

Questions: Are those the whole set? or are we missing something?

Let us try with x = 4,

5 + 4 < 9

this is equivalent to

9 < 9         which is false, since 9 = 9

 

In conclusion: 4 is not in the solution set, but any number less than 4 is

 

 

 

Property #1: Transitive

 

 

If a < b    and   b < c   then  a < c

Examples:

· - 3 < 9     and    9 < 12     then –3 < 12

· -5 < -2    and    –2 < 0      then –5 < 0

If a, b belong to the Real numbers, then a is greater than b (written a > b) if and only if b is less than a; with symbols we write

 

a > b if and only if b < a

The symbols ≤ (“is less than or equal to”) and ≥ (“is greater than or equal to”) are defined as follows:

 

a ≤ b if and only if  either a < b or a = b

a ≥b if and only if  either a > b or a = b

Inequalities

 

Definitions

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Properties of inequalities

Let a, b, c real numbers

Recognizing Solutions for Inequalities with Unknown Variables

Let          5 + x < 9,

we want to find the value (s) of x that results in a true expression.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Goal for Inequalities

Given any inequality that contains the unknown variable x, our goal is isolate x on one side of the inequality sign with a known quantity on the other side of the inequality sign.

Example: 5 + x < 9,

We know that 5 + x is less than 9, however, we want to know what is the inequality related with x alone. So our goal is take 5 + x < … and convert it in

x < …     or    …< x

Solving simple inequalities

We want to solve  5 + x < 9. Our goal is convert 5 + x < … in  x < … so we use the following steps

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The statements a < b, a > b, a ≤ b and a ≥b are called inequalities

Intervals are used to represent “solutions sets” of inequalities in one variable. The solution set of such an inequality is the set of all numbers that satisfy the inequality.

Property #2: Addition and subtraction

 

 

If a < b and c is a real number then,

a + c < b + c     and     a – c < b –  c 

Examples:

· -3 < 9      then      -3 + 6 < 9 + 6

· -3 < 9      then      -3 – 6 < 9 – 6

Property #3: Multiplication

 

 

If a < b and c is a negative real number then,

ca  > cb     

Examples:-3 is a negative number

 -3 < 9      then      (-3)(-3) > (-3)(9)

If a < b and c is a positive real number then

ca  < cb

Example: -3 < 9      then      3(-3) < 3(9)

Property #4: Division

 

 

If a < b and c is a negative real number then,

 

Example: -3 is a negative number

               -3 < 9      then  

Step 1: Find how to get rid of  the numerical term in this case 5, for a review click here

Step2: Use the property #2: subtract 5 on both sides of the inequality

–5 + 5 + x < –5 + 9

Or equivalently

x < 4

Step 3: The solution set. The solution set of the given inequality is the interval (–¥ , 4) 

Illustrate it on the real number line

 

 

 

 



Practice solving simple inequalities by following the step by step instructions below:



If a, b belong to the Real numbers, then a is less than b (written a < b) if and only if b - a is positive.

If a < b and c is a positive real number then

 

Example: -3 < 9      then