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Inequalities in Algebra


When learning and studying Algebra, one does need to become comfortable with the presence of inequalities in Algebra when they appear in expressions and equations.

Helpful are some properties of inequalities, which can be recalled and used in a range of situations where we may wish to simplify and solve for an inequality.



Inequalities Symbols:

At this point, we’re all used to seeing standard equalities in Math. Where something equals something else exactly on each side of an equals sign.

3 + 4 = 7 \space \space \space \space \space \space \space \space \space 5 \space {\text{--}} \space 2 = 3 \space \space \space \space \space \space \space \space \space 9 = 9



But we also have symbols to represent inequalities where something may not be exactly equal to something else.
Perhaps a value is less than or greater than another value.

The inequalities symbols we can encounter are as follows.

<  Less than.           <  Less than or equal to.

>  Greater than.      >  Greater than or equal to.


So instead of   3 + 4 = 7,

we could have    3 + 4 \lt 8,     as 7 is less than 8,     7 \lt 8.

Or    5 \space {\text{--}} \space 2 \gt 1,     as 3 is greater than 1,     3 \lt 1.


The “or equal to” inequality symbols become useful at times when variables are involved.
The properties of inequalities we show below do also apply to them also.







Inequalities in Algebra, Properties


We can perform operations on inequalities the same way that we perform operations on equalities.

Like with equalities though we must perform the operation on both sides on the inequality sign. Though we need to pay particular attention to division and multiplication, as we’ll see below.



Addition and Subtraction:

a < b     =>     a + c  <  b + c
a > b     =>     a + c  >  b + c

a < b     =>     a − c  <  b − c
a > b     =>     a − c  >  b − c


If we consider  5 \lt 8,   this is true.

Adding 2 to both sides.    5 + 2 \lt 8 + 2    =>    7 \lt 10
Still true.

Subtracting 2 from both sides.    5 \space {\text{--}} \space 2 \lt 8 \space {\text{--}} \space 2    =>    3 \lt 6
Still true.




Multiplication:

With multiplication and inequalities, we need to watch if the number multiplying is a positive number or a negative number.


If we consider   4 \lt 5,   this is true.

Multiplying both sides by  2.     4 \times 2 \lt 5 \times 2    =>    8 \lt 10
Still true.

Multiplying both sides by  –2.     4 \times {\text{-}}2 \lt 5 \times {\text{-}}2    =>    {\text{-}}8 \lt {\text{-}}10
NOT true.


10  is NOT greater than  –8,  so we have to change things slightly.

Which is to flip the inequality sign when a negative number is involved in the multiplication.

4 \times {\text{-}}2 \lt 5 \times {\text{-}}2    =>    {\text{-}}8 \gt {\text{-}}10

a < b     =>     a × c  <  b × c
a > b     =>     a × c  >  b × c

a < b     =>     a × -c  >  b × -c
a > b     =>     a × -c  >  b × -c




Division:

With division and properties of inequalities, we need to watch if the number dividing is a positive number or a negative number.


If we consider   6 \lt 8,   this is true.

Dividing both sides by  2.     \frac{6}{2} \lt \frac{8}{2}    =>    3 \lt 4
Still true.

Dividing both sides by  –2.     \frac{6}{{\text{-}}2} \lt \frac{8}{{\text{-}}2}    =>    {\text{-}}3 \lt {\text{-}}4
NOT true.


4  is NOT greater than  –3,  so like with multiplication, we have to change things slightly.

Which is again to flip the inequality sign, when the division involves a negative number.

\frac{6}{{\text{-}}2} \lt \frac{8}{{\text{-}}2}    =>    {\text{-}}3 \gt {\text{-}}4

a < b     =>     a ÷ c  <  b × c
a > b     =>     a ÷ c  >  b × c

a < b     =>     a ÷ -c  >  b × -c
a > b     =>     a ÷ -c  >  b × -c





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