An inequality is simply a comparison of two quantities or expressions.
a < b
a is less than b
a < b
a is less than or equal to b
a > b
a is greater than b
a > b
a is greater than or equal to b
5
Greater than
4
3
2
1
0
-1
-2
-3
-4
-5
Less than
If we set up a number line, you can see how numbers are set up for greater than or less than questions.
For example, 4 is greater than -4.
Inequalities and Algebra
The inequality 3x + 2 > x - 6 is solved just as an algebraic equation is solved.
3x + 2 > x - 6
3x > x - 8
Subtract 2 from each side
2x > - 8
Subtract x from each side.
x > -4
Divide by 2. Any number greater than -4 satisfies the inequality.
There are several rules that we must follow when manipulating inequalities:
The same number or algebraic expression may be added or subtracted from each side of an inequality.
The same positive number (or positive algebraic expression) may multiply or divide each side of an inequality.
Both sides of the same type of inequality may be added and the inequality remains.
(If x < y and w < z, then x + w < y + z).
If a negative number (or negative algebraic expression) multiplies or divides each side of an inequality, the inequality sign must be reversed. (Be sure to remember this; it often leads to errors!)
Example
Solve the inequality 2x - 2 > x - 5.
Solution
2x - 2 > x - 5
2x > x - 3
Add 2 to each side
x > - 3
Subtract x from each side.
Example
Solve the inequality 3r + 5 > 6r - 7.
Solution
3r + 5 > 6r - 7
3r > 6r - 12
Subtract 5 from each side.
-3r > -12
Subtract 6r from each side.
r < 4
Divide each side by (-3) and reverse the inequality symbol.
(NOTE: You must reverse the inequality symbol because you are dividing by a negative number).
