Inequalities

Inequalities show the relationship between two expressions that are not equal to one another. Inequalities are useful when projecting profits and breakeven figures.

Inequalities are the relationships between two expressions which are not equal to one another. Symbols for inequalities such as <, >, ≤, ≥ are used.

Example:

• 7 > x read as “7 is greater than x” or “x is less than 7”. 
•  x ≤ -4 read as “x is less than or equal to -4” or “-4 is greater than or equal to x”.

Inequalities on a number line:

Open circles are used for numbers that are less than or greater than (< or >). Closed circles are used for numbers that are less than or equal to and greater than or equal to (≤ or ≥).

For example, this is the number line for the inequality, x ≥ 0:

The symbol used is greater than or equal to (≥) so a closed circle must be used at 0. x
is greater than or equal to 0, so the arrow from the circle must show the numbers that are larger than 0.

Another example: 

Show the inequality y < 2 on a number line:

y is less than (<) 2, which means an open circle at 2 must be used. y is less than 2, so an arrow below the values of 2 must be drawn in.

Example 1:

 

Example 2:

Solve: 2x - 5 < 12

Solution:

2x - 5 < 12
(2x - 5) + 5 < 12 + 5
2x < 17
(1/2)2x < (1/2)17
x < 17/2




The solution set of the inequality is {x.:.x.<.17/2} which is read as "the set of all x such that x is less than 17/2".

 

Example 3:

Solve: 14(x-2) 132 - 281x

Solution:

14(x-2) 132 - 281x
14x - 28 132 - 281x
14x 160 - 281x
295x 160
x 160/295
x 32/59




The solution set is {x : x 32/59}.

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